Riemann-Siegel function Z: Series representations (formula 10.04.06.0014)

RiemannSiegelZ

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelZ[z]

Series representations

Generalized power series

Expansions at z==-i/2

http://functions.wolfram.com/10.04.06.0014.01

Input Form

RiemannSiegelZ[z] \[Proportional] (-(1/Sqrt[-2 I (z + I/2)])) (1 + (I/2) (EulerGamma - Log[2 Pi]) (z + I/2) + (1/48) (Pi^2 - 6 (EulerGamma - 3 Log[2 Pi]) (EulerGamma + Log[2 Pi]) + 48 Derivative[2][Zeta][0]) (z + I/2)^2 + (I/96) (-10 EulerGamma^3 - 18 EulerGamma^2 Log[2 Pi] - EulerGamma (Pi^2 + 6 (Log[2 Pi]^2 - 8 StieltjesGamma[1])) + Log[2 Pi] (Pi^2 + 2 Log[2 Pi]^2 + 48 StieltjesGamma[1]) - 96 StieltjesGamma[2] + 16 Zeta[3]) (z + I/2)^3 + \[Ellipsis]) /; (z -> -(I/2))

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02