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A254331 Decimal expansion of gamma_1(1/3), the first generalized Stieltjes constant at 1/3 (negated). 10
3, 2, 5, 9, 5, 5, 7, 5, 1, 5, 9, 1, 7, 9, 1, 0, 1, 9, 5, 2, 5, 0, 8, 7, 4, 5, 8, 2, 6, 7, 6, 5, 5, 9, 2, 5, 7, 9, 7, 6, 4, 7, 2, 2, 0, 4, 3, 9, 9, 4, 3, 0, 0, 4, 8, 1, 1, 7, 9, 7, 4, 8, 6, 7, 3, 8, 9, 7, 9, 3, 7, 0, 1, 4, 9, 5, 4, 6, 8, 7, 9, 2, 4, 7, 8, 9, 6, 5, 2, 5, 8, 8, 2, 0, 0, 8, 6, 7, 3, 8, 0, 4, 3, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
FORMULA
Equals integral_[0..infinity] (3*(-2*arctan(3*x) + 3*x*log(1/9 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 9*x^2)) dx - 3*log(3)/2 - log(3)^2/2.
EXAMPLE
-3.25955751591791019525087458267655925797647220439943...
MAPLE
evalf(int((3*(-2*arctan(3*x)+3*x*log(1/9+x^2)))/((-1+exp(2*Pi*x))*(9*x^2+1)), x = 0..infinity)-3*log(3)*(1/2)-(1/2)*log(3)^2, 120); # Vaclav Kotesovec, Jan 29 2015
MATHEMATICA
gamma1[1/3] = (1/6)*((-Sqrt[3])*Pi*(EulerGamma + Log[(24*Pi^(5/2))/Gamma[1/6]^3]) - 3*(Log[3]^2 + EulerGamma*Log[27] + 2*Log[3]*Log[2*Pi] + 2*Log[2*Pi]^2 + Log[3/(4*Pi^2)]*Log[6*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[1/3], 10, 104] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/3], 10, 104] // First
CROSSREFS
Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).
Sequence in context: A280875 A173701 A116627 * A210742 A175056 A320274
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved

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Last modified July 20 13:32 EDT 2024. Contains 374445 sequences. (Running on oeis4.)