# Zeros of the Riemann zeta function

Symbol: RiemannZeta $\zeta\!\left(s\right)$ Riemann zeta function
Symbol: RiemannZetaZero $\rho_{n}$ Nontrivial zero of the Riemann zeta function
Symbol: RiemannHypothesis $\operatorname{RH}$ Riemann hypothesis

## Main properties

$\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}(s) = \frac{1}{2} \;\text{ for all } s \in \mathbb{C} \text{ with } 0 \le \operatorname{Re}(s) \le 1 \;\mathbin{\operatorname{and}}\; \zeta\!\left(s\right) = 0\right)$
$\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{R}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 \le \operatorname{Re}(s) \le 1} \zeta\!\left(s\right) = \left\{ \rho_{n} : n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\} \cup \left\{ \rho_{n} : n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \right\}$
$0 < \operatorname{Re}\!\left(\rho_{n}\right) < 1$
$\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}$
$\rho_{-n} = \overline{\rho_{n}}$

## Numerical values

$\rho_{1} \in \frac{1}{2} + \left[14.134725141734693790457251983562470270784257115699 \pm 2.44 \cdot 10^{-49}\right] i$
$\rho_{2} \in \frac{1}{2} + \left[21.022039638771554992628479593896902777334340524903 \pm 2.19 \cdot 10^{-49}\right] i$
Table of $\operatorname{Im}\!\left(\rho_{n}\right)$ to 50 digits for $1 \le n \le 50$
Table of $\operatorname{Im}\!\left(\rho_{n}\right)$ to 10 digits for $1 \le n \le 500$
Table of $\operatorname{Im}\!\left(\rho_{{10}^{n}}\right)$ to 50 digits for $0 \le n \le 16$

Related topics: Riemann zeta function

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

2020-04-08 16:14:44.404316 UTC