Fungrim home page

Zeros of the Riemann zeta function

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

e0a6a2
Symbol: RiemannZeta ζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
669509
Symbol: RiemannZetaZero ρn\rho_{n} Nontrivial zero of the Riemann zeta function
c03de4
Symbol: RiemannHypothesis RH\operatorname{RH} Riemann hypothesis

Main properties

See also: Riemann hypothesis
9fa2a1
(RH)    (Re(s)=12   for all sC with 0Re(s)1  and  ζ ⁣(s)=0)\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}(s) = \frac{1}{2} \;\text{ for all } s \in \mathbb{C} \text{ with } 0 \le \operatorname{Re}(s) \le 1 \;\mathbin{\operatorname{and}}\; \zeta\!\left(s\right) = 0\right)
49704a
(RH)    (Re ⁣(ρn)=12   for all nZ1)\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)
2e1ff3
zerossRζ ⁣(s)={2n:nZ1}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{R}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}
a78abc
zerossC,0Re(s)1ζ ⁣(s)={ρn:nZ  and  n0}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 \le \operatorname{Re}(s) \le 1} \zeta\!\left(s\right) = \left\{ \rho_{n} : n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \right\}
692e42
zerossCζ ⁣(s)={2n:nZ1}{ρn:nZ  and  n0}\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\}
cbbf16
0<Re ⁣(ρn)<10 < \operatorname{Re}\!\left(\rho_{n}\right) < 1
e6ff64
Re ⁣(ρn)=12\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}
60c2ec
ρn=ρn\rho_{-n} = \overline{\rho_{n}}

Numerical values

945fa5
ρ112+[14.134725141734693790457251983562470270784257115699±2.441049]i\rho_{1} \in \frac{1}{2} + \left[14.134725141734693790457251983562470270784257115699 \pm 2.44 \cdot 10^{-49}\right] i
c0ae99
ρ212+[21.022039638771554992628479593896902777334340524903±2.191049]i\rho_{2} \in \frac{1}{2} + \left[21.022039638771554992628479593896902777334340524903 \pm 2.19 \cdot 10^{-49}\right] i
71d9d9
Table of Im ⁣(ρn)\operatorname{Im}\!\left(\rho_{n}\right) to 50 digits for 1n501 \le n \le 50
dc558b
Table of Im ⁣(ρn)\operatorname{Im}\!\left(\rho_{n}\right) to 10 digits for 1n5001 \le n \le 500
2e1cc7
Table of Im ⁣(ρ10n)\operatorname{Im}\!\left(\rho_{{10}^{n}}\right) to 50 digits for 0n160 \le n \le 16

Related topics: Riemann zeta function

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC