Ramanujan tau L function: Representations through equivalent functions (formula 10.10.27.0001)

RamanujanTauL

$Mathematica Notation$

Zeta Functions and Polylogarithms

RamanujanTauL[z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/10.10.27.0001.01

Input Form

RamanujanTauL[z] == (((2 Pi)^(-6 + z) RamanujanTauZ[I (6 - z)])/ Sqrt[Gamma[z]]) E^((1/2) (-Log[Gamma[12 - z]] + Log[Gamma[z]] + LogGamma[12 - z] - LogGamma[z])) Sqrt[Gamma[1 - z] ((11 - 12 z + z^2) (20 - 12 z + z^2) (27 - 12 z + z^2) (32 - 12 z + z^2) (35 - 12 z + z^2) (6 - z))]

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> <mi> τL </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 6 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> τZ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 12 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 12 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 20 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 27 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 32 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 35 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> τL </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -6 </cn> </apply> </apply> <apply> <ci> τZ </ci> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 12 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> </apply> <apply> <ci> LogGamma </ci> <apply> <plus /> <cn type='integer'> 12 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LogGamma </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 11 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 20 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 27 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 32 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 35 </cn> </apply> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamanujanTauL", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List[RowBox[List["-", "6"]], "+", "z"]]], " ", RowBox[List["RamanujanTauZ", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["6", "-", "z"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Gamma", "[", RowBox[List["12", "-", "z"]], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Gamma", "[", "z", "]"]], "]"]], "+", RowBox[List["LogGamma", "[", RowBox[List["12", "-", "z"]], "]"]], "-", RowBox[List["LogGamma", "[", "z", "]"]]]], ")"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["11", "-", RowBox[List["12", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["20", "-", RowBox[List["12", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["27", "-", RowBox[List["12", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["32", "-", RowBox[List["12", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["12", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["6", "-", "z"]], ")"]]]], ")"]]]]]]], SqrtBox[RowBox[List["Gamma", "[", "z", "]"]]]]]]]]

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

2007-05-02