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

Main properties

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\}
692e42
zerossCζ ⁣(s)={2n:nZ1}{ρn:nZandn0}\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 \lt \operatorname{Re}\!\left(\rho_{n}\right) \lt 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.

2019-05-23 08:00:13.607731 UTC