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 RiemannSiegelZ

 http://functions.wolfram.com/10.04.06.0008.01

 Input Form

 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, -1 + k}] 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])) - (1/(32 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]))))/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

 Standard Form

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 MathML Form

 Z TagBox["Z", RiemannSiegelZ] ( z ) ( - 1 ) k 2 - 2 k - 1 - 1 2 logΓ ( k + 1 2 ) π 1 4 - k ( 2 k ) ! ζ ( 2 k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] ( j = 0 k - 1 2 1 + 4 j - 2 z ) - 2 ( z + 2 + 2 k ) ( 1 + 4 ζ ( 2 k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] ( ( 4 ψ TagBox["\[Psi]", PolyGamma] ( 2 k + 1 ) + TagBox["\[DoubledGamma]", Function[EulerGamma]] - 2 log ( 4 π ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) ) ζ ( 2 k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] + 4 ζ ( 2 k + 1 ) ) ( z + 2 + 2 k ) - 1 32 ζ ( 2 k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] ( ζ ( 2 k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] ( 4 log 2 ( π ) - 16 log ( 2 π ) log ( π ) + 4 TagBox["\[DoubledGamma]", Function[EulerGamma]] log ( π ) + TagBox["\[DoubledGamma]", Function[EulerGamma]] 2 - π 2 + 16 log 2 ( 2 π ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) 2 + 16 ψ TagBox["\[Psi]", PolyGamma] ( 2 k + 1 ) 2 - 8 TagBox["\[DoubledGamma]", Function[EulerGamma]] log ( 2 π ) + 8 ( - log ( 16 ) - 2 log ( π ) + TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ψ TagBox["\[Psi]", PolyGamma] ( 2 k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) ( log ( 256 ) + 4 log ( π ) - 8 ψ TagBox["\[Psi]", PolyGamma] ( 2 k + 1 ) - 2 TagBox["\[DoubledGamma]", Function[EulerGamma]] ) + 16 ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( 2 k + 1 ) - 2 ζ ( 2 , k + 1 2 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["2", Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[\$CellContext`e1, \$CellContext`e2], Zeta[\$CellContext`e1, \$CellContext`e2]]]] ) + 8 ( - log ( 16 ) - 2 log ( π ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) + 4 ψ TagBox["\[Psi]", PolyGamma] ( 2 k + 1 ) + TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ζ ( 2 k + 1 ) + 16 ζ ′′ ( 2 k + 1 ) ) ( z + 2 + 2 k ) 2 + O ( ( z + 2 + 2 k ) 3 ) ) /; ( z "\[Rule]" - 2 - 2 k ) k + Condition Proportional RiemannSiegelZ z -1 k 2 -2 k -1 -1 1 2 LogGamma k 1 2 1 4 -1 k 2 k Zeta 2 k 1 j 0 k -1 2 1 4 j -1 2 z 1 2 -1 -2 z 2 -1 2 k 1 2 1 4 Zeta 2 k 1 -1 4 PolyGamma 2 k 1 -1 2 4 -1 PolyGamma k 1 2 Zeta 2 k 1 4 D Zeta 2 k 1 2 k 1 z 2 -1 2 k -1 1 32 Zeta 2 k 1 -1 Zeta 2 k 1 4 2 -1 16 2 4 2 -1 2 16 2 2 PolyGamma k 1 2 2 16 PolyGamma 2 k 1 2 -1 8 2 8 -1 16 -1 2 PolyGamma 2 k 1 PolyGamma k 1 2 256 4 -1 8 PolyGamma 2 k 1 -1 2 16 PolyGamma 1 2 k 1 -1 2 Zeta 2 k 1 2 8 -1 16 -1 2 -1 PolyGamma k 1 2 4 PolyGamma 2 k 1 D Zeta 2 k 1 2 k 1 16 D Zeta 2 k 1 2 k 1 2 z 2 -1 2 k 2 O z 2 -1 2 k 3 Rule z -1 2 -1 -1 2 k k SuperPlus [/itex]

 Rule Form

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

 2001-10-29