Riemann zeta function: Differentiation (formula 10.01.20.0010)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Differentiation

Low-order differentiation

Derivatives at zero

http://functions.wolfram.com/10.01.20.0010.01

Input Form

Derivative[5][Zeta][0] == (1/96) (192 EulerGamma^5 + 720 EulerGamma^4 Log[2 Pi] - 19 Pi^4 Log[2 Pi] + 80 EulerGamma^3 (Pi^2 + 12 (Log[2 Pi]^2 + StieltjesGamma[1])) + 240 EulerGamma (Pi^2 StieltjesGamma[1] + 4 (3 Log[2 Pi]^2 StieltjesGamma[1] + 3 Log[2 Pi] StieltjesGamma[2] + StieltjesGamma[3])) - 40 Pi^2 (Log[2 Pi]^3 - 6 Log[2 Pi] StieltjesGamma[1] - 3 StieltjesGamma[2] + 2 Zeta[3]) + 120 EulerGamma^2 (Pi^2 Log[2 Pi] + 4 (Log[2 Pi]^3 + 6 Log[2 Pi] StieltjesGamma[1] + 3 StieltjesGamma[2] + 2 Zeta[3])) - 48 (Log[2 Pi]^5 - 30 Log[2 Pi]^2 StieltjesGamma[2] - 20 Log[2 Pi] StieltjesGamma[3] - 5 StieltjesGamma[4] + 20 Log[2 Pi]^2 Zeta[3] - 20 StieltjesGamma[1] (Log[2 Pi]^3 + 2 Zeta[3]) + 24 Zeta[5]))

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "5", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 96 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 80 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> <mo> + </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 240 </mn> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> 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</mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mi> log </mi> <mn> 5 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 19 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 720 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> 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</apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 240 </cn> <eulergamma /> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn 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</cn> <apply> <plus /> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 720 </cn> <apply> <power /> <eulergamma /> <cn type='integer'> 4 </cn> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 192 </cn> <apply> <power /> <eulergamma /> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

Zeta[s,a]