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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Zeros > Sums over zeros





http://functions.wolfram.com/10.01.30.0006.01









  


  










Input Form





Subscript[Ζ,6]2*StieltjesGamma[1]^3 + 6*EulerGamma^4* StieltjesGamma[1] + StieltjesGamma[3]* StieltjesGamma[1] + EulerGamma^6 + (3*StieltjesGamma[2]^2)/4 + 3*EulerGamma^3* StieltjesGamma[2] + EulerGamma^2* (9*StieltjesGamma[1]^2 + StieltjesGamma[3]) + (1/4)*EulerGamma*(24*StieltjesGamma[1]* StieltjesGamma[2] + StieltjesGamma[4]) + StieltjesGamma[5]/20 - Pi^6/960 + 1/; Subscript[Ζ,6]Limit[Sum[1/Subscript[ρ,k]^6,Abs[Subscript[ρ,k]<T]], TInfinity]&&Zeta[Subscript[ρ,k]]0&& Re[Subscript[ρ,k]]≠0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "6"], "\[Equal]", RowBox[List["1", "+", SuperscriptBox["EulerGamma", "6"], "-", FractionBox[SuperscriptBox["\[Pi]", "6"], "960"], "+", RowBox[List["6", " ", SuperscriptBox["EulerGamma", "4"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "3"]]], "+", RowBox[List["3", " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "2", "]"]], "2"]]], "4"], "+", RowBox[List[RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "+", RowBox[List[SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["9", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "2"]]], "+", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["24", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["StieltjesGamma", "[", "4", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["StieltjesGamma", "[", "5", "]"]], "20"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "6"], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[UnderscriptBox["\[Sum]", RowBox[List[RowBox[List["\[LeftBracketingBar]", SubscriptBox["\[Rho]", "k"], "\[RightBracketingBar]"]], "<", "T"]]], FractionBox["1", SubsuperscriptBox["\[Rho]", "k", "6"]]]], ",", RowBox[List["T", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]]]]]]]]










MathML Form







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</ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 9 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#947; </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#947; </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> TagBox </ci> <ms> &#8509; </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 24 </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#947; 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Date Added to functions.wolfram.com (modification date)





2003-08-21





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