RiemannSiegelZ[z] \[Proportional] ((-1)^k 2^(-(1/2) - 2 k) Pi^(1/4 - k) Sqrt[I (-(I/2) - 2 I k + z)] (2 k)! Product[2/Sqrt[1 + 4 j + 2 I z], {j, 0, k - 1}] Zeta[1 + 2 k] (1 - (1/(4 Zeta[1 + 2 k])) (I (-(I/2) - 2 I k + z) ((EulerGamma - 2 Log[4 Pi] - PolyGamma[1/2 + k] + 4 PolyGamma[1 + 2 k]) Zeta[1 + 2 k] + 4 Derivative[1][Zeta][1 + 2 k])) - ((-(I/2) - 2 I k + z)^2 (Zeta[1 + 2 k] (EulerGamma^2 - Pi^2 + 4 EulerGamma Log[Pi] + 4 Log[Pi]^2 - 8 EulerGamma Log[2 Pi] - 16 Log[Pi] Log[2 Pi] + 16 Log[2 Pi]^2 + PolyGamma[1/2 + k]^2 + PolyGamma[1/2 + k] (-2 EulerGamma + Log[256] + 4 Log[Pi] - 8 PolyGamma[1 + 2 k]) + 8 (EulerGamma - Log[16] - 2 Log[Pi]) PolyGamma[1 + 2 k] + 16 PolyGamma[1 + 2 k]^2 + 16 PolyGamma[1, 1 + 2 k] - 2 Zeta[2, 1/2 + k]) + 8 (EulerGamma - Log[16] - 2 Log[Pi] - PolyGamma[1/2 + k] + 4 PolyGamma[1 + 2 k]) Derivative[1][Zeta][1 + 2 k] + 16 Derivative[2][Zeta][1 + 2 k]))/(32 Zeta[1 + 2 k])))/ E^((1/2) LogGamma[1/2 + k]) + O[(-(I/2) - 2 I k + z)^3] /; (z -> I/2 + 2 I k) && Element[k, Integers] && k > 0

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2001-10-29