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Fungrim entry: 9ee8bc

ζ ⁣(s)=2(2π)s1sin ⁣(πs2)Γ ⁣(1s)ζ ⁣(1s)\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:sC  and  sZ0s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}
Alternative assumptions:sC[[x]]  and  sZ0s \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
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Sinsin(z)\sin(z) Sine
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PowerSeriesK[[x]]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)))))

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2020-04-08 16:14:44.404316 UTC