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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Differentiation > Low-order differentiation > Derivatives at zero





http://functions.wolfram.com/10.01.20.0009.01









  


  










Input Form





Derivative[4][Zeta][0] == (3 EulerGamma^4)/2 - (19 Pi^4)/480 + 4 EulerGamma^3 Log[2 Pi] - (1/2) Log[2 Pi]^4 - (1/4) Pi^2 (Log[2 Pi]^2 - 2 StieltjesGamma[1]) + 6 Log[2 Pi]^2 StieltjesGamma[1] + EulerGamma^2 (Pi^2/4 + 3 Log[2 Pi]^2 + 6 StieltjesGamma[1]) + 6 Log[2 Pi] StieltjesGamma[2] + 6 EulerGamma (2 Log[2 Pi] StieltjesGamma[1] + StieltjesGamma[2]) + 2 StieltjesGamma[3] - 4 Log[2 Pi] Zeta[3]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", "4", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["3", " ", SuperscriptBox["EulerGamma", "4"]]], "2"], "-", FractionBox[RowBox[List["19", " ", SuperscriptBox["\[Pi]", "4"]]], "480"], "+", RowBox[List["4", " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "4"]]], "-", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List[SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"]]], "+", RowBox[List["6", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["6", " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Zeta", "[", "3", "]"]]]]]]]]]]










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





2007-05-02





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