# Fungrim entry: 9ee8bc

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}

s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Sin$\sin(z)$ Sine
Gamma$\Gamma(z)$ Gamma function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
PowerSeries$K[[x]]$ Formal power series
Source code for this entry:
Entry(ID("9ee8bc"),
Formula(Equal(RiemannZeta(s), Mul(Mul(Mul(Mul(2, Pow(Mul(2, Pi), Sub(s, 1))), Sin(Div(Mul(Pi, s), 2))), Gamma(Sub(1, s))), RiemannZeta(Sub(1, s))))),
Variables(s),
Assumptions(And(Element(s, CC), NotElement(s, ZZGreaterEqual(0))), And(Element(s, PowerSeries(CC, SerX)), NotElement(s, ZZGreaterEqual(0)))))

## 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