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http://functions.wolfram.com/10.02.20.0022.01
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Derivative[1, 0][Zeta][s, 1/2] == 2^s Log[2] Zeta[s] +
(2^s - 1) Derivative[1][Zeta][s]
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["s", ",", FractionBox["1", "2"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", "s"], " ", RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List["Zeta", "[", "s", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "s"], " ", "-", "1"]], ")"]], RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", "s", "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["s", Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mi> s </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <ci> s </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> s </ci> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["s_", ",", FractionBox["1", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", "s"], " ", RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List["Zeta", "[", "s", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "s"], "-", "1"]], ")"]], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", "s", "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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