Zeta[s, -(7/2)] == 2^s (1 + 3^(-s) + 5^(-s) + 7^(-s)) + (-1 + 2^s) Zeta[s]

Cell[BoxData[RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List["-", FractionBox["7", "2"]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", "s"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["3", RowBox[List["-", "s"]]], "+", SuperscriptBox["5", RowBox[List["-", "s"]]], "+", SuperscriptBox["7", RowBox[List["-", "s"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["2", "s"]]], ")"]], " ", RowBox[List["Zeta", "[", "s", "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;s&quot;, Zeta, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;7&quot;, &quot;2&quot;]]], Zeta, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> &#63449; </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mn> 3 </mn> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> + </mo> <msup> <mn> 5 </mn> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> + </mo> <msup> <mn> 7 </mn> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mn> 2 </mn> <mi> s </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;s&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", RowBox[List["s_", ",", RowBox[List["-", FractionBox["7", "2"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", "s"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["3", RowBox[List["-", "s"]]], "+", SuperscriptBox["5", RowBox[List["-", "s"]]], "+", SuperscriptBox["7", RowBox[List["-", "s"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["2", "s"]]], ")"]], " ", RowBox[List["Zeta", "[", "s", "]"]]]]]]]]]]

2007-05-02