Riemann zeta function: Differentiation (formula 10.01.20.0015)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Differentiation

Low-order differentiation

Derivatives at zero

http://functions.wolfram.com/10.01.20.0015.01

Input Form

Derivative[10][Zeta][0] == (9 EulerGamma^10)/2 - (95265 Pi^10)/22528 + 40 EulerGamma^9 Log[2 Pi] - (1/2) Log[2 Pi]^10 - (11813/512) Pi^8 (Log[2 Pi]^2 - 2 StieltjesGamma[1]) + 45 Log[2 Pi]^8 StieltjesGamma[1] + (15/8) EulerGamma^8 (7 Pi^2 + 84 Log[2 Pi]^2 + 24 StieltjesGamma[1]) + 180 Log[2 Pi]^7 StieltjesGamma[2] + 420 Log[2 Pi]^6 StieltjesGamma[3] + 630 Log[2 Pi]^5 StieltjesGamma[4] + 630 Log[2 Pi]^4 StieltjesGamma[5] + 420 Log[2 Pi]^3 StieltjesGamma[6] + 180 Log[2 Pi]^2 StieltjesGamma[7] + 45 Log[2 Pi] StieltjesGamma[8] + 5 StieltjesGamma[9] - 120 Log[2 Pi]^7 Zeta[3] + 5040 Log[2 Pi]^5 StieltjesGamma[1] Zeta[3] + 12600 Log[2 Pi]^4 StieltjesGamma[2] Zeta[3] + 16800 Log[2 Pi]^3 StieltjesGamma[3] Zeta[3] + 12600 Log[2 Pi]^2 StieltjesGamma[4] Zeta[3] + 5040 Log[2 Pi] StieltjesGamma[5] Zeta[3] + 840 StieltjesGamma[6] Zeta[3] - 4200 Log[2 Pi]^4 Zeta[3]^2 + 50400 Log[2 Pi]^2 StieltjesGamma[1] Zeta[3]^2 + 50400 Log[2 Pi] StieltjesGamma[2] Zeta[3]^2 + 16800 StieltjesGamma[3] Zeta[3]^2 - 11200 Log[2 Pi] Zeta[3]^3 + 90 EulerGamma^7 (Pi^2 Log[2 Pi] + 4 Log[2 Pi]^3 + 4 Log[2 Pi] StieltjesGamma[1] + 2 StieltjesGamma[2] + 8 Zeta[3]) - (1375/64) Pi^6 (Log[2 Pi]^4 - 12 Log[2 Pi]^2 StieltjesGamma[1] - 12 Log[2 Pi] StieltjesGamma[2] - 4 StieltjesGamma[3] + 8 Log[2 Pi] Zeta[3]) + (35/16) EulerGamma^6 (19 Pi^4 + 24 Pi^2 (5 Log[2 Pi]^2 + 2 StieltjesGamma[1]) + 48 (5 Log[2 Pi]^4 + 12 Log[2 Pi]^2 StieltjesGamma[1] + 12 Log[2 Pi] StieltjesGamma[2] + 4 StieltjesGamma[3] + 40 Log[2 Pi] Zeta[3])) - 3024 Log[2 Pi]^5 Zeta[5] + 60480 Log[2 Pi]^3 StieltjesGamma[1] Zeta[5] + 90720 Log[2 Pi]^2 StieltjesGamma[2] Zeta[5] + 60480 Log[2 Pi] StieltjesGamma[3] Zeta[5] + 15120 StieltjesGamma[4] Zeta[5] - 60480 Log[2 Pi]^2 Zeta[3] Zeta[5] + 120960 StieltjesGamma[1] Zeta[3] Zeta[5] - 36288 Zeta[5]^2 - (133/16) Pi^4 (Log[2 Pi]^6 - 60 Log[2 Pi]^2 StieltjesGamma[3] - 30 Log[2 Pi] StieltjesGamma[4] - 6 StieltjesGamma[5] + 40 Log[2 Pi]^3 Zeta[3] + 40 Zeta[3]^2 - 60 StieltjesGamma[2] (Log[2 Pi]^3 + 2 Zeta[3]) - 30 StieltjesGamma[1] (Log[2 Pi]^4 + 8 Log[2 Pi] Zeta[3]) + 144 Log[2 Pi] Zeta[5]) + (21/2) EulerGamma^5 (19 Pi^4 Log[2 Pi] + 10 Pi^2 (4 Log[2 Pi]^3 + 6 Log[2 Pi] StieltjesGamma[1] + 3 StieltjesGamma[2] + 8 Zeta[3]) + 12 (4 Log[2 Pi]^5 + 30 Log[2 Pi]^2 StieltjesGamma[2] + 20 Log[2 Pi] StieltjesGamma[3] + 5 StieltjesGamma[4] + 80 Log[2 Pi]^2 Zeta[3] + 20 StieltjesGamma[1] (Log[2 Pi]^3 + 2 Zeta[3]) + 96 Zeta[5])) + (15/64) EulerGamma^4 (275 Pi^6 + 532 Pi^4 (3 Log[2 Pi]^2 + 2 StieltjesGamma[1]) + 560 Pi^2 (3 Log[2 Pi]^4 + 12 Log[2 Pi]^2 StieltjesGamma[1] + 12 Log[2 Pi] StieltjesGamma[2] + 4 StieltjesGamma[3] + 24 Log[2 Pi] Zeta[3]) + 1344 (Log[2 Pi]^6 + 20 Log[2 Pi]^2 StieltjesGamma[3] + 10 Log[2 Pi] StieltjesGamma[4] + 2 StieltjesGamma[5] + 40 Log[2 Pi]^3 Zeta[3] + 40 Zeta[3]^2 + 20 StieltjesGamma[2] (Log[2 Pi]^3 + 2 Zeta[3]) + 10 StieltjesGamma[1] (Log[2 Pi]^4 + 8 Log[2 Pi] Zeta[3]) + 144 Log[2 Pi] Zeta[5])) - 43200 Log[2 Pi]^3 Zeta[7] + 259200 Log[2 Pi] StieltjesGamma[1] Zeta[7] + 129600 StieltjesGamma[2] Zeta[7] - 86400 Zeta[3] Zeta[7] - (15/8) Pi^2 (Log[2 Pi]^8 - 280 Log[2 Pi]^4 StieltjesGamma[3] - 280 Log[2 Pi]^3 StieltjesGamma[4] - 168 Log[2 Pi]^2 StieltjesGamma[5] - 56 Log[2 Pi] StieltjesGamma[6] - 8 StieltjesGamma[7] + 112 Log[2 Pi]^5 Zeta[3] - 2240 Log[2 Pi] StieltjesGamma[3] Zeta[3] - 560 StieltjesGamma[4] Zeta[3] + 1120 Log[2 Pi]^2 Zeta[3]^2 + 1344 Log[2 Pi]^3 Zeta[5] + 2688 Zeta[3] Zeta[5] - 168 StieltjesGamma[2] (Log[2 Pi]^5 + 20 Log[2 Pi]^2 Zeta[3] + 24 Zeta[5]) - 56 StieltjesGamma[1] (Log[2 Pi]^6 + 40 Log[2 Pi]^3 Zeta[3] + 40 Zeta[3]^2 + 144 Log[2 Pi] Zeta[5]) + 5760 Log[2 Pi] Zeta[7]) + (5/8) EulerGamma^3 (275 Pi^6 Log[2 Pi] + 266 Pi^4 (2 Log[2 Pi]^3 + 6 Log[2 Pi] StieltjesGamma[1] + 3 StieltjesGamma[2] + 4 Zeta[3]) + 168 Pi^2 (2 Log[2 Pi]^5 + 30 Log[2 Pi]^2 StieltjesGamma[2] + 20 Log[2 Pi] StieltjesGamma[3] + 5 StieltjesGamma[4] + 40 Log[2 Pi]^2 Zeta[3] + 20 StieltjesGamma[1] (Log[2 Pi]^3 + 2 Zeta[3]) + 48 Zeta[5]) + 96 (2 Log[2 Pi]^7 + 140 Log[2 Pi]^3 StieltjesGamma[3] + 105 Log[2 Pi]^2 StieltjesGamma[4] + 42 Log[2 Pi] StieltjesGamma[5] + 7 StieltjesGamma[6] + 140 Log[2 Pi]^4 Zeta[3] + 280 StieltjesGamma[3] Zeta[3] + 560 Log[2 Pi] Zeta[3]^2 + 105 StieltjesGamma[2] (Log[2 Pi]^4 + 8 Log[2 Pi] Zeta[3]) + 1008 Log[2 Pi]^2 Zeta[5] + 42 StieltjesGamma[1] (Log[2 Pi]^5 + 20 Log[2 Pi]^2 Zeta[3] + 24 Zeta[5]) + 1440 Zeta[7])) + (1/512) EulerGamma^2 (11813 Pi^8 + 66000 Pi^6 (Log[2 Pi]^2 + 2 StieltjesGamma[1]) + 63840 Pi^4 (Log[2 Pi]^4 + 12 Log[2 Pi]^2 StieltjesGamma[1] + 12 Log[2 Pi] StieltjesGamma[2] + 4 StieltjesGamma[3] + 8 Log[2 Pi] Zeta[3]) + 26880 Pi^2 (Log[2 Pi]^6 + 60 Log[2 Pi]^2 StieltjesGamma[3] + 30 Log[2 Pi] StieltjesGamma[4] + 6 StieltjesGamma[5] + 40 Log[2 Pi]^3 Zeta[3] + 40 Zeta[3]^2 + 60 StieltjesGamma[2] (Log[2 Pi]^3 + 2 Zeta[3]) + 30 StieltjesGamma[1] (Log[2 Pi]^4 + 8 Log[2 Pi] Zeta[3]) + 144 Log[2 Pi] Zeta[5]) + 11520 (Log[2 Pi]^8 + 280 Log[2 Pi]^4 StieltjesGamma[3] + 280 Log[2 Pi]^3 StieltjesGamma[4] + 168 Log[2 Pi]^2 StieltjesGamma[5] + 56 Log[2 Pi] StieltjesGamma[6] + 8 StieltjesGamma[7] + 112 Log[2 Pi]^5 Zeta[3] + 2240 Log[2 Pi] StieltjesGamma[3] Zeta[3] + 560 StieltjesGamma[4] Zeta[3] + 1120 Log[2 Pi]^2 Zeta[3]^2 + 1344 Log[2 Pi]^3 Zeta[5] + 2688 Zeta[3] Zeta[5] + 168 StieltjesGamma[2] (Log[2 Pi]^5 + 20 Log[2 Pi]^2 Zeta[3] + 24 Zeta[5]) + 56 StieltjesGamma[1] (Log[2 Pi]^6 + 40 Log[2 Pi]^3 Zeta[3] + 40 Zeta[3]^2 + 144 Log[2 Pi] Zeta[5]) + 5760 Log[2 Pi] Zeta[7])) + (15/16) EulerGamma (275 Pi^6 (2 Log[2 Pi] StieltjesGamma[1] + StieltjesGamma[2]) + 266 Pi^4 (6 Log[2 Pi]^2 StieltjesGamma[2] + 4 Log[2 Pi] StieltjesGamma[3] + StieltjesGamma[4] + 4 StieltjesGamma[1] (Log[2 Pi]^3 + 2 Zeta[3])) + 112 Pi^2 (20 Log[2 Pi]^3 StieltjesGamma[3] + 15 Log[2 Pi]^2 StieltjesGamma[4] + 6 Log[2 Pi] StieltjesGamma[5] + StieltjesGamma[6] + 40 StieltjesGamma[3] Zeta[3] + 15 StieltjesGamma[2] (Log[2 Pi]^4 + 8 Log[2 Pi] Zeta[3]) + 6 StieltjesGamma[1] (Log[2 Pi]^5 + 20 Log[2 Pi]^2 Zeta[3] + 24 Zeta[5])) + 48 (56 Log[2 Pi]^5 StieltjesGamma[3] + 70 Log[2 Pi]^4 StieltjesGamma[4] + 56 Log[2 Pi]^3 StieltjesGamma[5] + 28 Log[2 Pi]^2 StieltjesGamma[6] + 8 Log[2 Pi] StieltjesGamma[7] + StieltjesGamma[8] + 1120 Log[2 Pi]^2 StieltjesGamma[3] Zeta[3] + 560 Log[2 Pi] StieltjesGamma[4] Zeta[3] + 112 StieltjesGamma[5] Zeta[3] + 1344 StieltjesGamma[3] Zeta[5] + 28 StieltjesGamma[2] (Log[2 Pi]^6 + 40 Log[2 Pi]^3 Zeta[3] + 40 Zeta[3]^2 + 144 Log[2 Pi] Zeta[5]) + 8 StieltjesGamma[1] (Log[2 Pi]^7 + 70 Log[2 Pi]^4 Zeta[3] + 280 Log[2 Pi] Zeta[3]^2 + 504 Log[2 Pi]^2 Zeta[5] + 720 Zeta[7]))) - 201600 Log[2 Pi] Zeta[9]

Standard Form

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MathML Form

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12 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mi> log </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 35 </mn> <mn> 16 </mn> </mfrac> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mn> 6 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) 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