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Fungrim entry: 1a63af

ζ ⁣(1s)=2cos ⁣(12πs)(2π)sΓ(s)ζ ⁣(s)\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:sC  and  s{1,0,}s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}
Alternative assumptions:sC[[x]]  and  s{1,0,}s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}
\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\}
Fungrim symbol Notation Short description
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
PowerSeriesK[[x]]K[[x]] Formal power series
Source code for this entry:
    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)))),
    Assumptions(And(Element(s, CC), NotElement(s, ZZLessEqual(1))), And(Element(s, PowerSeries(CC, SerX)), NotElement(s, ZZLessEqual(1)))))

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2021-03-15 19:12:00.328586 UTC