Zeros of the Riemann zeta function

Table of contents: Main properties - Numerical values - Related topics

Symbol: RiemannZeta —

\zeta\!\left(s\right)

— Riemann zeta function

The Riemann zeta function

\zeta\!\left(s\right)

is a function of one complex variable

s

. It is a meromorphic function with a pole at

s = 1

. The following table lists all conditions such that RiemannZeta(s) is defined in Fungrim.

Domain	Codomain
Numbers
$s \in \left(1, \infty\right)$	$\zeta\!\left(s\right) \in \left(1, \infty\right)$
$s \in \mathbb{R} \setminus \left\{1\right\}$	$\zeta\!\left(s\right) \in \mathbb{R}$
$s \in \mathbb{C} \setminus \left\{1\right\}$	$\zeta\!\left(s\right) \in \mathbb{C}$
Infinities
$s \in \left\{1\right\}$	$\zeta\!\left(s\right) \in \left\{{\tilde \infty}\right\}$
$s \in \left\{\infty\right\}$	$\zeta\!\left(s\right) \in \left\{1\right\}$
Formal power series
$s \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{R}[[x]]$
$s \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{C}[[x]]$
$s \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{R}(\!(x)\!)$
$s \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{C}(\!(x)\!)$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`FormalPowerSeries`	$K[[x]]$	Formal power series
`FormalLaurentSeries`	$K(\!(x)\!)$	Formal Laurent series

Source code for this entry:

Entry(ID("e0a6a2"),
    SymbolDefinition(RiemannZeta, RiemannZeta(s), "Riemann zeta function"),
    Description("The Riemann zeta function", RiemannZeta(s), "is a function of one complex variable", s, ". It is a meromorphic function with a pole at", Equal(s, 1), ".", "The following table lists all conditions such that", SourceForm(RiemannZeta(s)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(s, OpenInterval(1, Infinity)), Element(RiemannZeta(s), OpenInterval(1, Infinity))), Tuple(Element(s, SetMinus(RR, Set(1))), Element(RiemannZeta(s), RR)), Tuple(Element(s, SetMinus(CC, Set(1))), Element(RiemannZeta(s), CC)), TableSection("Infinities"), Tuple(Element(s, Set(1)), Element(RiemannZeta(s), Set(UnsignedInfinity))), Tuple(Element(s, Set(Infinity)), Element(RiemannZeta(s), Set(1))), TableSection("Formal power series"), Tuple(And(Element(s, FormalPowerSeries(RR, x)), Unequal(SeriesCoefficient(s, x, 0), 1)), Element(RiemannZeta(s), FormalPowerSeries(RR, x))), Tuple(And(Element(s, FormalPowerSeries(CC, x)), Unequal(SeriesCoefficient(s, x, 0), 1)), Element(RiemannZeta(s), FormalPowerSeries(CC, x))), Tuple(And(Element(s, FormalPowerSeries(RR, x)), Unequal(s, 1)), Element(RiemannZeta(s), FormalLaurentSeries(RR, x))), Tuple(And(Element(s, FormalPowerSeries(CC, x)), Unequal(s, 1)), Element(RiemannZeta(s), FormalLaurentSeries(CC, x))))))

669509

Symbol: RiemannZetaZero —

\rho_{n}

— Nontrivial zero of the Riemann zeta function

Domain	Codomain
$n \in \mathbb{Z} \setminus \left\{0\right\}$	$\rho_{n} \in \mathbb{C}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("669509"),
    SymbolDefinition(RiemannZetaZero, RiemannZetaZero(n), "Nontrivial zero of the Riemann zeta function"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(Element(n, SetMinus(ZZ, Set(0))), Element(RiemannZetaZero(n), CC)))))

Main properties

2e1ff3

\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{R}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{R}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`RR`	$\mathbb{R}$	Real numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("2e1ff3"),
    Formula(Equal(Zeros(RiemannZeta(s), s, Element(s, RR)), SetBuilder(Neg(Mul(2, n)), n, Element(n, ZZGreaterEqual(1))))))

692e42

\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\} \cup \left\{ \rho_{n} : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\} \cup \left\{ \rho_{n} : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("692e42"),
    Formula(Equal(Zeros(RiemannZeta(s), s, Element(s, CC)), Union(SetBuilder(Neg(Mul(2, n)), n, Element(n, ZZGreaterEqual(1))), SetBuilder(RiemannZetaZero(n), n, And(Element(n, ZZ), Unequal(n, 0)))))))

cbbf16

0 \lt \operatorname{Re}\!\left(\rho_{n}\right) \lt 1

Assumptions:

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0

TeX:

0 \lt \operatorname{Re}\!\left(\rho_{n}\right) \lt 1

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("cbbf16"),
    Formula(Less(0, Re(RiemannZetaZero(n)), 1)),
    Variables(n),
    Assumptions(And(Element(n, ZZ), Unequal(n, 0))))

e6ff64

\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}

Assumptions:

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \,\mathbin{\operatorname{and}}\, \left(\left|n\right| \lt 103800788359 \,\mathbin{\operatorname{or}}\, \operatorname{RiemannHypothesis}\right)

References:

D. J. Platt (2016), Isolating some non-trivial zeros of zeta, Mathematics of Computation 86(307):1, DOI: 10.1090/mcom/3198

TeX:

\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \,\mathbin{\operatorname{and}}\, \left(\left|n\right| \lt 103800788359 \,\mathbin{\operatorname{or}}\, \operatorname{RiemannHypothesis}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ZZ`	$\mathbb{Z}$	Integers
`Abs`	$\left\|z\right\|$	Absolute value
`RiemannHypothesis`	$\operatorname{RiemannHypothesis}$	Riemann hypothesis

Source code for this entry:

Entry(ID("e6ff64"),
    Formula(Equal(Re(RiemannZetaZero(n)), Div(1, 2))),
    Variables(n),
    Assumptions(And(Element(n, ZZ), Unequal(n, 0), Or(Less(Abs(n), 103800788359), RiemannHypothesis))),
    References("D. J. Platt (2016), Isolating some non-trivial zeros of zeta, Mathematics of Computation 86(307):1, DOI: 10.1090/mcom/3198"))

60c2ec

\rho_{-n} = \overline{\rho_{n}}

Assumptions:

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0

TeX:

\rho_{-n} = \overline{\rho_{n}}

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`Conjugate`	$\overline{z}$	Complex conjugate
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("60c2ec"),
    Formula(Equal(RiemannZetaZero(Neg(n)), Conjugate(RiemannZetaZero(n)))),
    Variables(n),
    Assumptions(And(Element(n, ZZ), Unequal(n, 0))))

Numerical values

945fa5

\rho_{1} \in \frac{1}{2} + \left[14.134725141734693790457251983562470270784257115699 \pm 2.44 \cdot 10^{-49}\right] i

TeX:

\rho_{1} \in \frac{1}{2} + \left[14.134725141734693790457251983562470270784257115699 \pm 2.44 \cdot 10^{-49}\right] i

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("945fa5"),
    Formula(Element(RiemannZetaZero(1), Add(Div(1, 2), Mul(RealBall(Decimal("14.134725141734693790457251983562470270784257115699"), Decimal("2.44e-49")), ConstI)))))

c0ae99

\rho_{2} \in \frac{1}{2} + \left[21.022039638771554992628479593896902777334340524903 \pm 2.19 \cdot 10^{-49}\right] i

TeX:

\rho_{2} \in \frac{1}{2} + \left[21.022039638771554992628479593896902777334340524903 \pm 2.19 \cdot 10^{-49}\right] i

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("c0ae99"),
    Formula(Element(RiemannZetaZero(2), Add(Div(1, 2), Mul(RealBall(Decimal("21.022039638771554992628479593896902777334340524903"), Decimal("2.19e-49")), ConstI)))))

71d9d9

Table of

\operatorname{Im}\!\left(\rho_{n}\right)

to 50 digits for

1 \le n \le 50

$n$	$\operatorname{Im}\!\left(\rho_{n}\right)$
1	14.134725141734693790457251983562470270784257115699
2	21.022039638771554992628479593896902777334340524903
3	25.010857580145688763213790992562821818659549672558
4	30.424876125859513210311897530584091320181560023715
5	32.935061587739189690662368964074903488812715603517
6	37.586178158825671257217763480705332821405597350831
7	40.918719012147495187398126914633254395726165962777
8	43.327073280914999519496122165406805782645668371837
9	48.005150881167159727942472749427516041686844001144
10	49.773832477672302181916784678563724057723178299677
11	52.970321477714460644147296608880990063825017888821
12	56.446247697063394804367759476706127552782264471717
13	59.347044002602353079653648674992219031098772806467
14	60.831778524609809844259901824524003802910090451219
15	65.112544048081606660875054253183705029348149295167
16	67.079810529494173714478828896522216770107144951746
17	69.546401711173979252926857526554738443012474209603
18	72.067157674481907582522107969826168390480906621457
19	75.704690699083933168326916762030345922811903530697
20	77.144840068874805372682664856304637015796032449234
21	79.337375020249367922763592877116228190613246743120
22	82.910380854086030183164837494770609497508880593782
23	84.735492980517050105735311206827741417106627934241
24	87.425274613125229406531667850919213252171886401269
25	88.809111207634465423682348079509378395444893409819
26	92.491899270558484296259725241810684878721794027731
27	94.651344040519886966597925815208153937728027015655
28	95.870634228245309758741029219246781695256461224988
29	98.831194218193692233324420138622327820658039063428
30	101.31785100573139122878544794029230890633286638430
31	103.72553804047833941639840810869528083448117306950
32	105.44662305232609449367083241411180899728275392854
33	107.16861118427640751512335196308619121347670788140
34	111.02953554316967452465645030994435041534596839007
35	111.87465917699263708561207871677059496031174987339
36	114.32022091545271276589093727619107980991765772383
37	116.22668032085755438216080431206475512732985123238
38	118.79078286597621732297913970269982434730621059281
39	121.37012500242064591894553297049992272300131063173
40	122.94682929355258820081746033077001649621438987386
41	124.25681855434576718473200796612992444157353877469
42	127.51668387959649512427932376690607626808830988155
43	129.57870419995605098576803390617997360864095326466
44	131.08768853093265672356637246150134905920354750298
45	133.49773720299758645013049204264060766497417494390
46	134.75650975337387133132606415716973617839606861365
47	138.11604205453344320019155519028244785983527462415
48	139.73620895212138895045004652338246084679005256538
49	141.12370740402112376194035381847535509030066087975
50	143.11184580762063273940512386891392996623310243035

Table data:

\left(n, y\right)

such that

\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \,\mathbin{\operatorname{and}}\, \operatorname{NearestDecimal}\!\left(\operatorname{Im}\!\left(\rho_{n}\right), 50\right) = y

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("71d9d9"),
    Description("Table of", Im(RiemannZetaZero(n)), "to 50 digits for", LessEqual(1, n, 50)),
    Table(TableRelation(Tuple(n, y), And(Equal(Re(RiemannZetaZero(n)), Div(1, 2)), Equal(NearestDecimal(Im(RiemannZetaZero(n)), 50), y))), TableHeadings(n, Im(RiemannZetaZero(n))), TableSplit(1), List(Tuple(1, Decimal("14.134725141734693790457251983562470270784257115699")), Tuple(2, Decimal("21.022039638771554992628479593896902777334340524903")), Tuple(3, Decimal("25.010857580145688763213790992562821818659549672558")), Tuple(4, Decimal("30.424876125859513210311897530584091320181560023715")), Tuple(5, Decimal("32.935061587739189690662368964074903488812715603517")), Tuple(6, Decimal("37.586178158825671257217763480705332821405597350831")), Tuple(7, Decimal("40.918719012147495187398126914633254395726165962777")), Tuple(8, Decimal("43.327073280914999519496122165406805782645668371837")), Tuple(9, Decimal("48.005150881167159727942472749427516041686844001144")), Tuple(10, Decimal("49.773832477672302181916784678563724057723178299677")), Tuple(11, Decimal("52.970321477714460644147296608880990063825017888821")), Tuple(12, Decimal("56.446247697063394804367759476706127552782264471717")), Tuple(13, Decimal("59.347044002602353079653648674992219031098772806467")), Tuple(14, Decimal("60.831778524609809844259901824524003802910090451219")), Tuple(15, Decimal("65.112544048081606660875054253183705029348149295167")), Tuple(16, Decimal("67.079810529494173714478828896522216770107144951746")), Tuple(17, Decimal("69.546401711173979252926857526554738443012474209603")), Tuple(18, Decimal("72.067157674481907582522107969826168390480906621457")), Tuple(19, Decimal("75.704690699083933168326916762030345922811903530697")), Tuple(20, Decimal("77.144840068874805372682664856304637015796032449234")), Tuple(21, Decimal("79.337375020249367922763592877116228190613246743120")), Tuple(22, Decimal("82.910380854086030183164837494770609497508880593782")), Tuple(23, Decimal("84.735492980517050105735311206827741417106627934241")), Tuple(24, Decimal("87.425274613125229406531667850919213252171886401269")), Tuple(25, Decimal("88.809111207634465423682348079509378395444893409819")), Tuple(26, Decimal("92.491899270558484296259725241810684878721794027731")), Tuple(27, Decimal("94.651344040519886966597925815208153937728027015655")), Tuple(28, Decimal("95.870634228245309758741029219246781695256461224988")), Tuple(29, Decimal("98.831194218193692233324420138622327820658039063428")), Tuple(30, Decimal("101.31785100573139122878544794029230890633286638430")), Tuple(31, Decimal("103.72553804047833941639840810869528083448117306950")), Tuple(32, Decimal("105.44662305232609449367083241411180899728275392854")), Tuple(33, Decimal("107.16861118427640751512335196308619121347670788140")), Tuple(34, Decimal("111.02953554316967452465645030994435041534596839007")), Tuple(35, Decimal("111.87465917699263708561207871677059496031174987339")), Tuple(36, Decimal("114.32022091545271276589093727619107980991765772383")), Tuple(37, Decimal("116.22668032085755438216080431206475512732985123238")), Tuple(38, Decimal("118.79078286597621732297913970269982434730621059281")), Tuple(39, Decimal("121.37012500242064591894553297049992272300131063173")), Tuple(40, Decimal("122.94682929355258820081746033077001649621438987386")), Tuple(41, Decimal("124.25681855434576718473200796612992444157353877469")), Tuple(42, Decimal("127.51668387959649512427932376690607626808830988155")), Tuple(43, Decimal("129.57870419995605098576803390617997360864095326466")), Tuple(44, Decimal("131.08768853093265672356637246150134905920354750298")), Tuple(45, Decimal("133.49773720299758645013049204264060766497417494390")), Tuple(46, Decimal("134.75650975337387133132606415716973617839606861365")), Tuple(47, Decimal("138.11604205453344320019155519028244785983527462415")), Tuple(48, Decimal("139.73620895212138895045004652338246084679005256538")), Tuple(49, Decimal("141.12370740402112376194035381847535509030066087975")), Tuple(50, Decimal("143.11184580762063273940512386891392996623310243035")))))

dc558b

Table of

\operatorname{Im}\!\left(\rho_{n}\right)

to 10 digits for

1 \le n \le 500

$n$	$\operatorname{Im}\!\left(\rho_{n}\right)$
1	14.13472514
2	21.02203964
3	25.01085758
4	30.42487613
5	32.93506159
6	37.58617816
7	40.91871901
8	43.32707328
9	48.00515088
10	49.77383248
11	52.97032148
12	56.44624770
13	59.34704400
14	60.83177852
15	65.11254405
16	67.07981053
17	69.54640171
18	72.06715767
19	75.70469070
20	77.14484007
21	79.33737502
22	82.91038085
23	84.73549298
24	87.42527461
25	88.80911121
26	92.49189927
27	94.65134404
28	95.87063423
29	98.83119422
30	101.3178510
31	103.7255380
32	105.4466231
33	107.1686112
34	111.0295355
35	111.8746592
36	114.3202209
37	116.2266803
38	118.7907829
39	121.3701250
40	122.9468293
41	124.2568186
42	127.5166839
43	129.5787042
44	131.0876885
45	133.4977372
46	134.7565098
47	138.1160421
48	139.7362090
49	141.1237074
50	143.1118458
51	146.0009825
52	147.4227653
53	150.0535204
54	150.9252576
55	153.0246938
56	156.1129093
57	157.5975918
58	158.8499882
59	161.1889641
60	163.0307097
61	165.5370692
62	167.1844400
63	169.0945154
64	169.9119765
65	173.4115365
66	174.7541915
67	176.4414343
68	178.3774078
69	179.9164840
70	182.2070785
71	184.8744678
72	185.5987837
73	187.2289226
74	189.4161587
75	192.0266564
76	193.0797266
77	195.2653967
78	196.8764818
79	198.0153097
80	201.2647519
81	202.4935945
82	204.1896718
83	205.3946972
84	207.9062589
85	209.5765097
86	211.6908626
87	213.3479194
88	214.5470448
89	216.1695385
90	219.0675963
91	220.7149188
92	221.4307056
93	224.0070003
94	224.9833247
95	227.4214443
96	229.3374133
97	231.2501887
98	231.9872353
99	233.6934042
100	236.5242297

$n$	$\operatorname{Im}\!\left(\rho_{n}\right)$
101	237.7698205
102	239.5554776
103	241.0491578
104	242.8232719
105	244.0708985
106	247.1369901
107	248.1019901
108	249.5736896
109	251.0149478
110	253.0699867
111	255.3062565
112	256.3807137
113	258.6104395
114	259.8744070
115	260.8050845
116	263.5738939
117	265.5578518
118	266.6149738
119	267.9219151
120	269.9704490
121	271.4940556
122	273.4596092
123	275.5874926
124	276.4520495
125	278.2507435
126	279.2292509
127	282.4651148
128	283.2111857
129	284.8359640
130	286.6674454
131	287.9119205
132	289.5798549
133	291.8462913
134	293.5584341
135	294.9653696
136	295.5732549
137	297.9792771
138	299.8403261
139	301.6493255
140	302.6967496
141	304.8643713
142	305.7289126
143	307.2194961
144	310.1094631
145	311.1651415
146	312.4278012
147	313.9852857
148	315.4756161
149	317.7348059
150	318.8531043
151	321.1601343
152	322.1445587
153	323.4669696
154	324.8628661
155	327.4439013
156	329.0330717
157	329.9532397
158	331.4744676
159	333.6453785
160	334.2113548
161	336.8418504
162	338.3399929
163	339.8582167
164	341.0422611
165	342.0548775
166	344.6617029
167	346.3478706
168	347.2726776
169	349.3162609
170	350.4084193
171	351.8786490
172	353.4889005
173	356.0175750
174	357.1513023
175	357.9526851
176	359.7437550
177	361.2893617
178	363.3313306
179	364.7360241
180	366.2127103
181	367.9935755
182	368.9684381
183	370.0509192
184	373.0619284
185	373.8648739
186	375.8259128
187	376.3240922
188	378.4366802
189	379.8729753
190	381.4844686
191	383.4435294
192	384.9561168
193	385.8613008
194	387.2228902
195	388.8461284
196	391.4560836
197	392.2450833
198	393.4277438
199	395.5828700
200	396.3818542

$n$	$\operatorname{Im}\!\left(\rho_{n}\right)$
201	397.9187362
202	399.9851199
203	401.8392286
204	402.8619178
205	404.2364418
206	405.1343875
207	407.5814604
208	408.9472455
209	410.5138692
210	411.9722678
211	413.2627361
212	415.0188098
213	415.4552150
214	418.3877058
215	419.8613648
216	420.6438276
217	422.0767101
218	423.7165796
219	425.0698825
220	427.2088251
221	428.1279141
222	430.3287454
223	431.3013069
224	432.1386417
225	433.8892185
226	436.1610064
227	437.5816982
228	438.6217387
229	439.9184422
230	441.6831992
231	442.9045463
232	444.3193363
233	446.8606227
234	447.4417042
235	449.1485457
236	450.1269458
237	451.4033084
238	453.9867378
239	454.9746838
240	456.3284267
241	457.9038931
242	459.5134153
243	460.0879444
244	462.0653673
245	464.0572869
246	465.6715392
247	466.5702869
248	467.4390462
249	469.5360046
250	470.7736555
251	472.7991747
252	473.8352323
253	475.6003394
254	476.7690152
255	478.0752638
256	478.9421815
257	481.8303394
258	482.8347828
259	483.8514272
260	485.5391481
261	486.5287183
262	488.3805671
263	489.6617616
264	491.3988216
265	493.3144416
266	493.9579978
267	495.3588288
268	496.4296962
269	498.5807824
270	500.3090849
271	501.6044470
272	502.2762703
273	504.4997733
274	505.4152317
275	506.4641527
276	508.8007003
277	510.2642279
278	511.5622897
279	512.6231445
280	513.6689856
281	515.4350572
282	517.5896686
283	518.2342231
284	520.1063104
285	521.5251934
286	522.4566962
287	523.9605309
288	525.0773857
289	527.9036416
290	528.4062139
291	529.8062263
292	530.8669179
293	532.6881830
294	533.7796308
295	535.6643141
296	537.0697591
297	538.4285262
298	540.2131664
299	540.6313902
300	541.8474371

$n$	$\operatorname{Im}\!\left(\rho_{n}\right)$
301	544.3238901
302	545.6368332
303	547.0109121
304	547.9316134
305	549.4975676
306	550.9700100
307	552.0495722
308	553.7649721
309	555.7920206
310	556.8994764
311	557.5646592
312	559.3162370
313	560.2408075
314	562.5592076
315	564.1608791
316	564.5060559
317	566.6987877
318	567.7317579
319	568.9239552
320	570.0511148
321	572.4199841
322	573.6146105
323	575.0938860
324	575.8072471
325	577.0390035
326	579.0988347
327	580.1369594
328	581.9465763
329	583.2360882
330	584.5617059
331	585.9845632
332	586.7427719
333	588.1396633
334	590.6603975
335	591.7258581
336	592.5713583
337	593.9747147
338	595.7281537
339	596.3627683
340	598.4930773
341	599.5456404
342	601.6021367
343	602.5791679
344	603.6256189
345	604.6162185
346	606.3834604
347	608.4132173
348	609.3895752
349	610.8391629
350	611.7742096
351	613.5997787
352	614.6462379
353	615.5385634
354	618.1128314
355	619.1844826
356	620.2728937
357	621.7092945
358	622.3750027
359	624.2699000
360	626.0192834
361	627.2683969
362	628.3258624
363	630.4738874
364	630.8057809
365	632.2251412
366	633.5468583
367	635.5238003
368	637.3971932
369	637.9255140
370	638.9279383
371	640.6947947
372	641.9454997
373	643.2788838
374	644.9905782
375	646.3481916
376	647.7617530
377	648.7864009
378	650.1975193
379	650.6686839
380	653.6495716
381	654.3019206
382	655.7094630
383	656.9640846
384	658.1756144
385	659.6638460
386	660.7167326
387	662.2965864
388	664.2446047
389	665.3427631
390	666.5151477
391	667.1484949
392	668.9758488
393	670.3235852
394	672.4581836
395	673.0435783
396	674.3558978
397	676.1396744
398	677.2301807
399	677.8004447
400	679.7421979

$n$	$\operatorname{Im}\!\left(\rho_{n}\right)$
401	681.8949915
402	682.6027350
403	684.0135498
404	684.9726299
405	686.1632236
406	687.9615432
407	689.3689414
408	690.4747350
409	692.4516844
410	693.1769701
411	694.5339087
412	695.7263359
413	696.6260699
414	699.1320955
415	700.2967391
416	701.3017430
417	702.2273431
418	704.0338393
419	705.1258140
420	706.1846548
421	708.2690709
422	709.2295886
423	711.1302742
424	711.9002899
425	712.7493835
426	714.0827718
427	716.1123965
428	717.4825697
429	718.7427865
430	719.6971010
431	721.3511622
432	722.2775050
433	723.8458210
434	724.5626139
435	727.0564032
436	728.4054816
437	728.7587498
438	730.4164821
439	731.4173549
440	732.8180527
441	734.7896433
442	735.7654592
443	737.0529289
444	738.5804212
445	739.9095237
446	740.5738074
447	741.7573356
448	743.8950131
449	745.3449896
450	746.4993059
451	747.6745636
452	748.2427545
453	750.6559504
454	750.9663811
455	752.8876216
456	754.3223705
457	755.8393090
458	756.7682484
459	758.1017292
460	758.9002382
461	760.2823670
462	762.7000332
463	763.5930662
464	764.3075227
465	766.0875401
466	767.2184722
467	768.2814618
468	769.6934073
469	771.0708393
470	772.9616176
471	774.1177446
472	775.0478471
473	775.9997120
474	777.2997485
475	779.1570769
476	780.3489250
477	782.1376644
478	782.5979439
479	784.2888226
480	785.7390897
481	786.4611475
482	787.4684638
483	790.0590924
484	790.8316205
485	792.4277076
486	792.8886526
487	794.4837919
488	795.6065962
489	797.2634700
490	798.7075702
491	799.6543362
492	801.6042465
493	802.5419849
494	803.2430962
495	804.7622391
496	805.8616357
497	808.1518149
498	809.1977834
499	810.0818049
500	811.1843588

Table data:

\left(n, y\right)

such that

\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \,\mathbin{\operatorname{and}}\, \operatorname{NearestDecimal}\!\left(\operatorname{Im}\!\left(\rho_{n}\right), 10\right) = y

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("dc558b"),
    Description("Table of", Im(RiemannZetaZero(n)), "to 10 digits for", LessEqual(1, n, 500)),
    Table(TableRelation(Tuple(n, y), And(Equal(Re(RiemannZetaZero(n)), Div(1, 2)), Equal(NearestDecimal(Im(RiemannZetaZero(n)), 10), y))), TableHeadings(n, Im(RiemannZetaZero(n))), TableSplit(5), List(Tuple(1, Decimal("14.13472514")), Tuple(2, Decimal("21.02203964")), Tuple(3, Decimal("25.01085758")), Tuple(4, Decimal("30.42487613")), Tuple(5, Decimal("32.93506159")), Tuple(6, Decimal("37.58617816")), Tuple(7, Decimal("40.91871901")), Tuple(8, Decimal("43.32707328")), Tuple(9, Decimal("48.00515088")), Tuple(10, Decimal("49.77383248")), Tuple(11, Decimal("52.97032148")), Tuple(12, Decimal("56.44624770")), Tuple(13, Decimal("59.34704400")), Tuple(14, Decimal("60.83177852")), Tuple(15, Decimal("65.11254405")), Tuple(16, Decimal("67.07981053")), Tuple(17, Decimal("69.54640171")), Tuple(18, Decimal("72.06715767")), Tuple(19, Decimal("75.70469070")), Tuple(20, Decimal("77.14484007")), Tuple(21, Decimal("79.33737502")), Tuple(22, Decimal("82.91038085")), Tuple(23, Decimal("84.73549298")), Tuple(24, Decimal("87.42527461")), Tuple(25, Decimal("88.80911121")), Tuple(26, Decimal("92.49189927")), Tuple(27, Decimal("94.65134404")), Tuple(28, Decimal("95.87063423")), Tuple(29, Decimal("98.83119422")), Tuple(30, Decimal("101.3178510")), Tuple(31, Decimal("103.7255380")), Tuple(32, Decimal("105.4466231")), Tuple(33, Decimal("107.1686112")), Tuple(34, Decimal("111.0295355")), Tuple(35, Decimal("111.8746592")), Tuple(36, Decimal("114.3202209")), Tuple(37, Decimal("116.2266803")), Tuple(38, Decimal("118.7907829")), Tuple(39, Decimal("121.3701250")), Tuple(40, Decimal("122.9468293")), Tuple(41, Decimal("124.2568186")), Tuple(42, Decimal("127.5166839")), Tuple(43, Decimal("129.5787042")), Tuple(44, Decimal("131.0876885")), Tuple(45, Decimal("133.4977372")), Tuple(46, Decimal("134.7565098")), Tuple(47, Decimal("138.1160421")), Tuple(48, Decimal("139.7362090")), Tuple(49, Decimal("141.1237074")), Tuple(50, Decimal("143.1118458")), Tuple(51, Decimal("146.0009825")), Tuple(52, Decimal("147.4227653")), Tuple(53, Decimal("150.0535204")), Tuple(54, Decimal("150.9252576")), Tuple(55, Decimal("153.0246938")), Tuple(56, Decimal("156.1129093")), Tuple(57, Decimal("157.5975918")), Tuple(58, Decimal("158.8499882")), Tuple(59, Decimal("161.1889641")), Tuple(60, Decimal("163.0307097")), Tuple(61, Decimal("165.5370692")), Tuple(62, Decimal("167.1844400")), Tuple(63, Decimal("169.0945154")), Tuple(64, Decimal("169.9119765")), Tuple(65, Decimal("173.4115365")), Tuple(66, Decimal("174.7541915")), Tuple(67, Decimal("176.4414343")), Tuple(68, Decimal("178.3774078")), Tuple(69, Decimal("179.9164840")), Tuple(70, Decimal("182.2070785")), Tuple(71, Decimal("184.8744678")), Tuple(72, Decimal("185.5987837")), Tuple(73, Decimal("187.2289226")), Tuple(74, Decimal("189.4161587")), Tuple(75, Decimal("192.0266564")), Tuple(76, Decimal("193.0797266")), Tuple(77, Decimal("195.2653967")), Tuple(78, Decimal("196.8764818")), Tuple(79, Decimal("198.0153097")), Tuple(80, Decimal("201.2647519")), Tuple(81, Decimal("202.4935945")), Tuple(82, Decimal("204.1896718")), Tuple(83, Decimal("205.3946972")), Tuple(84, Decimal("207.9062589")), Tuple(85, Decimal("209.5765097")), Tuple(86, Decimal("211.6908626")), Tuple(87, Decimal("213.3479194")), Tuple(88, Decimal("214.5470448")), Tuple(89, Decimal("216.1695385")), Tuple(90, Decimal("219.0675963")), Tuple(91, Decimal("220.7149188")), Tuple(92, Decimal("221.4307056")), Tuple(93, Decimal("224.0070003")), Tuple(94, Decimal("224.9833247")), Tuple(95, Decimal("227.4214443")), Tuple(96, Decimal("229.3374133")), Tuple(97, Decimal("231.2501887")), Tuple(98, Decimal("231.9872353")), Tuple(99, Decimal("233.6934042")), Tuple(100, Decimal("236.5242297")), Tuple(101, Decimal("237.7698205")), Tuple(102, Decimal("239.5554776")), Tuple(103, Decimal("241.0491578")), Tuple(104, Decimal("242.8232719")), Tuple(105, Decimal("244.0708985")), Tuple(106, Decimal("247.1369901")), Tuple(107, Decimal("248.1019901")), Tuple(108, Decimal("249.5736896")), Tuple(109, Decimal("251.0149478")), Tuple(110, Decimal("253.0699867")), Tuple(111, Decimal("255.3062565")), Tuple(112, Decimal("256.3807137")), Tuple(113, Decimal("258.6104395")), Tuple(114, Decimal("259.8744070")), Tuple(115, Decimal("260.8050845")), Tuple(116, Decimal("263.5738939")), Tuple(117, Decimal("265.5578518")), Tuple(118, Decimal("266.6149738")), Tuple(119, Decimal("267.9219151")), Tuple(120, Decimal("269.9704490")), Tuple(121, Decimal("271.4940556")), Tuple(122, Decimal("273.4596092")), Tuple(123, Decimal("275.5874926")), Tuple(124, Decimal("276.4520495")), Tuple(125, Decimal("278.2507435")), Tuple(126, Decimal("279.2292509")), Tuple(127, Decimal("282.4651148")), Tuple(128, Decimal("283.2111857")), Tuple(129, Decimal("284.8359640")), Tuple(130, Decimal("286.6674454")), Tuple(131, Decimal("287.9119205")), Tuple(132, Decimal("289.5798549")), Tuple(133, Decimal("291.8462913")), Tuple(134, Decimal("293.5584341")), Tuple(135, Decimal("294.9653696")), Tuple(136, Decimal("295.5732549")), Tuple(137, Decimal("297.9792771")), Tuple(138, Decimal("299.8403261")), Tuple(139, Decimal("301.6493255")), Tuple(140, Decimal("302.6967496")), Tuple(141, Decimal("304.8643713")), Tuple(142, Decimal("305.7289126")), Tuple(143, Decimal("307.2194961")), Tuple(144, Decimal("310.1094631")), Tuple(145, Decimal("311.1651415")), Tuple(146, Decimal("312.4278012")), Tuple(147, Decimal("313.9852857")), Tuple(148, Decimal("315.4756161")), Tuple(149, Decimal("317.7348059")), Tuple(150, Decimal("318.8531043")), Tuple(151, Decimal("321.1601343")), Tuple(152, Decimal("322.1445587")), Tuple(153, Decimal("323.4669696")), Tuple(154, Decimal("324.8628661")), Tuple(155, Decimal("327.4439013")), Tuple(156, Decimal("329.0330717")), Tuple(157, Decimal("329.9532397")), Tuple(158, Decimal("331.4744676")), Tuple(159, Decimal("333.6453785")), Tuple(160, Decimal("334.2113548")), Tuple(161, Decimal("336.8418504")), Tuple(162, Decimal("338.3399929")), Tuple(163, Decimal("339.8582167")), Tuple(164, Decimal("341.0422611")), Tuple(165, Decimal("342.0548775")), Tuple(166, Decimal("344.6617029")), Tuple(167, Decimal("346.3478706")), Tuple(168, Decimal("347.2726776")), Tuple(169, Decimal("349.3162609")), Tuple(170, Decimal("350.4084193")), Tuple(171, Decimal("351.8786490")), Tuple(172, Decimal("353.4889005")), Tuple(173, Decimal("356.0175750")), Tuple(174, Decimal("357.1513023")), Tuple(175, Decimal("357.9526851")), Tuple(176, Decimal("359.7437550")), Tuple(177, Decimal("361.2893617")), Tuple(178, Decimal("363.3313306")), Tuple(179, Decimal("364.7360241")), Tuple(180, Decimal("366.2127103")), Tuple(181, Decimal("367.9935755")), Tuple(182, Decimal("368.9684381")), Tuple(183, Decimal("370.0509192")), Tuple(184, Decimal("373.0619284")), Tuple(185, Decimal("373.8648739")), Tuple(186, Decimal("375.8259128")), Tuple(187, Decimal("376.3240922")), Tuple(188, Decimal("378.4366802")), Tuple(189, Decimal("379.8729753")), Tuple(190, Decimal("381.4844686")), Tuple(191, Decimal("383.4435294")), Tuple(192, Decimal("384.9561168")), Tuple(193, Decimal("385.8613008")), Tuple(194, Decimal("387.2228902")), Tuple(195, Decimal("388.8461284")), Tuple(196, Decimal("391.4560836")), Tuple(197, Decimal("392.2450833")), Tuple(198, Decimal("393.4277438")), Tuple(199, Decimal("395.5828700")), Tuple(200, Decimal("396.3818542")), Tuple(201, Decimal("397.9187362")), Tuple(202, Decimal("399.9851199")), Tuple(203, Decimal("401.8392286")), Tuple(204, Decimal("402.8619178")), Tuple(205, Decimal("404.2364418")), Tuple(206, Decimal("405.1343875")), Tuple(207, Decimal("407.5814604")), Tuple(208, Decimal("408.9472455")), Tuple(209, Decimal("410.5138692")), Tuple(210, Decimal("411.9722678")), Tuple(211, Decimal("413.2627361")), Tuple(212, Decimal("415.0188098")), Tuple(213, Decimal("415.4552150")), Tuple(214, Decimal("418.3877058")), Tuple(215, Decimal("419.8613648")), Tuple(216, Decimal("420.6438276")), Tuple(217, Decimal("422.0767101")), Tuple(218, Decimal("423.7165796")), Tuple(219, Decimal("425.0698825")), Tuple(220, Decimal("427.2088251")), Tuple(221, Decimal("428.1279141")), Tuple(222, Decimal("430.3287454")), Tuple(223, Decimal("431.3013069")), Tuple(224, Decimal("432.1386417")), Tuple(225, Decimal("433.8892185")), Tuple(226, Decimal("436.1610064")), Tuple(227, Decimal("437.5816982")), Tuple(228, Decimal("438.6217387")), Tuple(229, Decimal("439.9184422")), Tuple(230, Decimal("441.6831992")), Tuple(231, Decimal("442.9045463")), Tuple(232, Decimal("444.3193363")), Tuple(233, Decimal("446.8606227")), Tuple(234, Decimal("447.4417042")), Tuple(235, Decimal("449.1485457")), Tuple(236, Decimal("450.1269458")), Tuple(237, Decimal("451.4033084")), Tuple(238, Decimal("453.9867378")), Tuple(239, Decimal("454.9746838")), Tuple(240, Decimal("456.3284267")), Tuple(241, Decimal("457.9038931")), Tuple(242, Decimal("459.5134153")), Tuple(243, Decimal("460.0879444")), Tuple(244, Decimal("462.0653673")), Tuple(245, Decimal("464.0572869")), Tuple(246, Decimal("465.6715392")), Tuple(247, Decimal("466.5702869")), Tuple(248, Decimal("467.4390462")), Tuple(249, Decimal("469.5360046")), Tuple(250, Decimal("470.7736555")), Tuple(251, Decimal("472.7991747")), Tuple(252, Decimal("473.8352323")), Tuple(253, Decimal("475.6003394")), Tuple(254, Decimal("476.7690152")), Tuple(255, Decimal("478.0752638")), Tuple(256, Decimal("478.9421815")), Tuple(257, Decimal("481.8303394")), Tuple(258, Decimal("482.8347828")), Tuple(259, Decimal("483.8514272")), Tuple(260, Decimal("485.5391481")), Tuple(261, Decimal("486.5287183")), Tuple(262, Decimal("488.3805671")), Tuple(263, Decimal("489.6617616")), Tuple(264, Decimal("491.3988216")), Tuple(265, Decimal("493.3144416")), Tuple(266, Decimal("493.9579978")), Tuple(267, Decimal("495.3588288")), Tuple(268, Decimal("496.4296962")), Tuple(269, Decimal("498.5807824")), Tuple(270, Decimal("500.3090849")), Tuple(271, Decimal("501.6044470")), Tuple(272, Decimal("502.2762703")), Tuple(273, Decimal("504.4997733")), Tuple(274, Decimal("505.4152317")), Tuple(275, Decimal("506.4641527")), Tuple(276, Decimal("508.8007003")), Tuple(277, Decimal("510.2642279")), Tuple(278, Decimal("511.5622897")), Tuple(279, Decimal("512.6231445")), Tuple(280, Decimal("513.6689856")), Tuple(281, Decimal("515.4350572")), Tuple(282, Decimal("517.5896686")), Tuple(283, Decimal("518.2342231")), Tuple(284, Decimal("520.1063104")), Tuple(285, Decimal("521.5251934")), Tuple(286, Decimal("522.4566962")), Tuple(287, Decimal("523.9605309")), Tuple(288, Decimal("525.0773857")), Tuple(289, Decimal("527.9036416")), Tuple(290, Decimal("528.4062139")), Tuple(291, Decimal("529.8062263")), Tuple(292, Decimal("530.8669179")), Tuple(293, Decimal("532.6881830")), Tuple(294, Decimal("533.7796308")), Tuple(295, Decimal("535.6643141")), Tuple(296, Decimal("537.0697591")), Tuple(297, Decimal("538.4285262")), Tuple(298, Decimal("540.2131664")), Tuple(299, Decimal("540.6313902")), Tuple(300, Decimal("541.8474371")), Tuple(301, Decimal("544.3238901")), Tuple(302, Decimal("545.6368332")), Tuple(303, Decimal("547.0109121")), Tuple(304, Decimal("547.9316134")), Tuple(305, Decimal("549.4975676")), Tuple(306, Decimal("550.9700100")), Tuple(307, Decimal("552.0495722")), Tuple(308, Decimal("553.7649721")), Tuple(309, Decimal("555.7920206")), Tuple(310, Decimal("556.8994764")), Tuple(311, Decimal("557.5646592")), Tuple(312, Decimal("559.3162370")), Tuple(313, Decimal("560.2408075")), Tuple(314, Decimal("562.5592076")), Tuple(315, Decimal("564.1608791")), Tuple(316, Decimal("564.5060559")), Tuple(317, Decimal("566.6987877")), Tuple(318, Decimal("567.7317579")), Tuple(319, Decimal("568.9239552")), Tuple(320, Decimal("570.0511148")), Tuple(321, Decimal("572.4199841")), Tuple(322, Decimal("573.6146105")), Tuple(323, Decimal("575.0938860")), Tuple(324, Decimal("575.8072471")), Tuple(325, Decimal("577.0390035")), Tuple(326, Decimal("579.0988347")), Tuple(327, Decimal("580.1369594")), Tuple(328, 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Tuple(356, Decimal("620.2728937")), Tuple(357, Decimal("621.7092945")), Tuple(358, Decimal("622.3750027")), Tuple(359, Decimal("624.2699000")), Tuple(360, Decimal("626.0192834")), Tuple(361, Decimal("627.2683969")), Tuple(362, Decimal("628.3258624")), Tuple(363, Decimal("630.4738874")), Tuple(364, Decimal("630.8057809")), Tuple(365, Decimal("632.2251412")), Tuple(366, Decimal("633.5468583")), Tuple(367, Decimal("635.5238003")), Tuple(368, Decimal("637.3971932")), Tuple(369, Decimal("637.9255140")), Tuple(370, Decimal("638.9279383")), Tuple(371, Decimal("640.6947947")), Tuple(372, Decimal("641.9454997")), Tuple(373, Decimal("643.2788838")), Tuple(374, Decimal("644.9905782")), Tuple(375, Decimal("646.3481916")), Tuple(376, Decimal("647.7617530")), Tuple(377, Decimal("648.7864009")), Tuple(378, Decimal("650.1975193")), Tuple(379, Decimal("650.6686839")), Tuple(380, Decimal("653.6495716")), Tuple(381, Decimal("654.3019206")), Tuple(382, Decimal("655.7094630")), Tuple(383, Decimal("656.9640846")), Tuple(384, Decimal("658.1756144")), Tuple(385, Decimal("659.6638460")), Tuple(386, Decimal("660.7167326")), Tuple(387, Decimal("662.2965864")), Tuple(388, Decimal("664.2446047")), Tuple(389, Decimal("665.3427631")), Tuple(390, Decimal("666.5151477")), Tuple(391, Decimal("667.1484949")), Tuple(392, Decimal("668.9758488")), Tuple(393, Decimal("670.3235852")), Tuple(394, Decimal("672.4581836")), Tuple(395, Decimal("673.0435783")), Tuple(396, Decimal("674.3558978")), Tuple(397, Decimal("676.1396744")), Tuple(398, Decimal("677.2301807")), Tuple(399, Decimal("677.8004447")), Tuple(400, Decimal("679.7421979")), Tuple(401, Decimal("681.8949915")), Tuple(402, Decimal("682.6027350")), Tuple(403, Decimal("684.0135498")), Tuple(404, Decimal("684.9726299")), Tuple(405, Decimal("686.1632236")), Tuple(406, Decimal("687.9615432")), Tuple(407, Decimal("689.3689414")), Tuple(408, Decimal("690.4747350")), Tuple(409, Decimal("692.4516844")), Tuple(410, Decimal("693.1769701")), 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Tuple(466, Decimal("767.2184722")), Tuple(467, Decimal("768.2814618")), Tuple(468, Decimal("769.6934073")), Tuple(469, Decimal("771.0708393")), Tuple(470, Decimal("772.9616176")), Tuple(471, Decimal("774.1177446")), Tuple(472, Decimal("775.0478471")), Tuple(473, Decimal("775.9997120")), Tuple(474, Decimal("777.2997485")), Tuple(475, Decimal("779.1570769")), Tuple(476, Decimal("780.3489250")), Tuple(477, Decimal("782.1376644")), Tuple(478, Decimal("782.5979439")), Tuple(479, Decimal("784.2888226")), Tuple(480, Decimal("785.7390897")), Tuple(481, Decimal("786.4611475")), Tuple(482, Decimal("787.4684638")), Tuple(483, Decimal("790.0590924")), Tuple(484, Decimal("790.8316205")), Tuple(485, Decimal("792.4277076")), Tuple(486, Decimal("792.8886526")), Tuple(487, Decimal("794.4837919")), Tuple(488, Decimal("795.6065962")), Tuple(489, Decimal("797.2634700")), Tuple(490, Decimal("798.7075702")), Tuple(491, Decimal("799.6543362")), Tuple(492, Decimal("801.6042465")), Tuple(493, Decimal("802.5419849")), Tuple(494, Decimal("803.2430962")), Tuple(495, Decimal("804.7622391")), Tuple(496, Decimal("805.8616357")), Tuple(497, Decimal("808.1518149")), Tuple(498, Decimal("809.1977834")), Tuple(499, Decimal("810.0818049")), Tuple(500, Decimal("811.1843588")))))

2e1cc7

Table of

\operatorname{Im}\!\left(\rho_{{10}^{n}}\right)

to 50 digits for

0 \le n \le 16

$n$	$\operatorname{Im}\!\left(\rho_{{10}^{n}}\right)$
0	14.134725141734693790457251983562470270784257115699
1	49.773832477672302181916784678563724057723178299677
2	236.52422966581620580247550795566297868952949521219
3	1419.4224809459956864659890380799168192321006010642
4	9877.7826540055011427740990706901235776224680517811
5	74920.827498994186793849200946918346620223555216802
6	600269.67701244495552123391427049074396819125790619
7	4992381.0140031786660182508391600932712387635814368
8	42653549.760951553903050309232819667982595130452178
9	371870203.83702805273405479598662519100082698522485
10	3293531632.3971367042089917031338769677069644102625
11	29538618431.613072810689561192671546108506486777642
12	267653395648.62594824214264940920070899588029633790
13	2445999556030.2468813938032396773514175248139258741
14	22514484222485.729124253904444090280880182979014905
15	208514052006405.46942460229754774510609948399247941
16	1941393531395154.7112809113883108073327538053720311

Table data:

\left(n, y\right)

such that

\operatorname{Re}\!\left(\rho_{{10}^{n}}\right) = \frac{1}{2} \,\mathbin{\operatorname{and}}\, \operatorname{NearestDecimal}\!\left(\operatorname{Im}\!\left(\rho_{{10}^{n}}\right), 50\right) = y

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`Pow`	${a}^{b}$	Power
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("2e1cc7"),
    Description("Table of", Im(RiemannZetaZero(Pow(10, n))), "to 50 digits for", LessEqual(0, n, 16)),
    Table(TableRelation(Tuple(n, y), And(Equal(Re(RiemannZetaZero(Pow(10, n))), Div(1, 2)), Equal(NearestDecimal(Im(RiemannZetaZero(Pow(10, n))), 50), y))), TableHeadings(n, Im(RiemannZetaZero(Pow(10, n)))), TableSplit(1), List(Tuple(0, Decimal("14.134725141734693790457251983562470270784257115699")), Tuple(1, Decimal("49.773832477672302181916784678563724057723178299677")), Tuple(2, Decimal("236.52422966581620580247550795566297868952949521219")), Tuple(3, Decimal("1419.4224809459956864659890380799168192321006010642")), Tuple(4, Decimal("9877.7826540055011427740990706901235776224680517811")), Tuple(5, Decimal("74920.827498994186793849200946918346620223555216802")), Tuple(6, Decimal("600269.67701244495552123391427049074396819125790619")), Tuple(7, Decimal("4992381.0140031786660182508391600932712387635814368")), Tuple(8, Decimal("42653549.760951553903050309232819667982595130452178")), Tuple(9, Decimal("371870203.83702805273405479598662519100082698522485")), Tuple(10, Decimal("3293531632.3971367042089917031338769677069644102625")), Tuple(11, Decimal("29538618431.613072810689561192671546108506486777642")), Tuple(12, Decimal("267653395648.62594824214264940920070899588029633790")), Tuple(13, Decimal("2445999556030.2468813938032396773514175248139258741")), Tuple(14, Decimal("22514484222485.729124253904444090280880182979014905")), Tuple(15, Decimal("208514052006405.46942460229754774510609948399247941")), Tuple(16, Decimal("1941393531395154.7112809113883108073327538053720311")))))