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RiemannSiegelZ






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > RiemannSiegelZ[z] > Series representations > Generalized power series > Expansions at z==i/2





http://functions.wolfram.com/10.04.06.0013.01









  


  










Input Form





RiemannSiegelZ[z] \[Proportional] (-(1/Sqrt[2 I (z - I/2)])) (1 - (I/2) (EulerGamma - Log[2 Pi]) (z - I/2) + (1/48) (Pi^2 - 6 (EulerGamma - 3 Log[2 Pi]) (EulerGamma + Log[2 Pi]) + 48 Derivative[2][Zeta][0]) (z - I/2)^2 - (I/96) (-10 EulerGamma^3 - 18 EulerGamma^2 Log[2 Pi] - EulerGamma (Pi^2 + 6 (Log[2 Pi]^2 - 8 StieltjesGamma[1])) + Log[2 Pi] (Pi^2 + 2 Log[2 Pi]^2 + 48 StieltjesGamma[1]) - 96 StieltjesGamma[2] + 16 Zeta[3]) (z - I/2)^3 + \[Ellipsis]) /; (z -> I/2)










Standard Form





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MathML Form







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&#960; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 96 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mfrac> <mi> &#8520; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> RiemannSiegelZ </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> 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/> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> 0 </cn> </apply> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 2 </cn> </list> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 96 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -18 </cn> <apply> <power /> <eulergamma /> <cn type='integer'> 2 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Rule Form





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





2007-05-02





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