Ramanujan tau zeta function: Representations through equivalent functions (formula 10.11.27.0002)

RamanujanTauZ

$Mathematica Notation$

Zeta Functions and Polylogarithms

RamanujanTauZ[z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/10.11.27.0002.01

Input Form

RamanujanTauZ[z] == ((E^((1/2) (-LogGamma[6 - I z] + LogGamma[6 + I z])) Sqrt[Gamma[6 - I z]])/((2 Pi)^(I z) Sqrt[I z (1 + z^2) (4 + z^2) (9 + z^2) (16 + z^2) (25 + z^2) Gamma[-5 - I z]])) RamanujanTauL[6 + I z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["RamanujanTauZ", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["LogGamma", "[", RowBox[List["6", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List["LogGamma", "[", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["6", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["9", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["16", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["25", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]], RowBox[List["RamanujanTauL", "[", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> τZ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 16 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 25 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mi> τL </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> τZ </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ci> LogGamma </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LogGamma </ci> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 16 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 25 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='integer'> -5 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> τL </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 6 </cn> </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