Riemann zeta function: Specific values (formula 10.01.03.0056)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Specific values

Values at fixed points

http://functions.wolfram.com/10.01.03.0056.01

Input Form

Zeta[3] == (-(2/3)) Pi^2 (6 Log[Glaisher] + Log[2 Pi] - 12 PolyGamma[-4, 1])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Zeta", "[", "3", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "-", RowBox[List["12", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "4"]], ",", "1"]], "]"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo>  </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox["A", Function[List[], Glaisher]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -4 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", "3", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "-", RowBox[List["12", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "4"]], ",", "1"]], "]"]]]]]], ")"]]]]]]]]

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

2007-05-02

Zeta[s,a]