# Fungrim entry: 1a63af

$\zeta\!\left(1 - s\right) = \frac{2 \cos\!\left(\frac{1}{2} \pi s\right)}{{\left(2 \pi\right)}^{s}} \Gamma(s) \zeta\!\left(s\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}$
Alternative assumptions:$s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}$
TeX:
\zeta\!\left(1 - s\right) = \frac{2 \cos\!\left(\frac{1}{2} \pi s\right)}{{\left(2 \pi\right)}^{s}} \Gamma(s) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}

s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}
Definitions:
Fungrim symbol Notation Short description
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
Gamma$\Gamma(z)$ Gamma function
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
PowerSeries$K[[x]]$ Formal power series
Source code for this entry:
Entry(ID("1a63af"),
Formula(Equal(RiemannZeta(Sub(1, s)), Mul(Mul(Div(Mul(2, Cos(Mul(Mul(Div(1, 2), Pi), s))), Pow(Mul(2, Pi), s)), Gamma(s)), RiemannZeta(s)))),
Variables(s),
Assumptions(And(Element(s, CC), NotElement(s, ZZLessEqual(1))), And(Element(s, PowerSeries(CC, SerX)), NotElement(s, ZZLessEqual(1)))))

## Topics using this entry

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