OFFSET
0,2
REFERENCES
Jonathan Borwein and Peter Borwein, Experimental and Computational Mathematics: Selected Writings, Perfectly Scientific Press, 2010, p. 106.
LINKS
D. J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998;
FORMULA
V = Sum_{j>k>0} (-1)^j cos(2Pi k/3)/(j^3 k).
Equals 3 zeta(3)/8-1/2+Sum_{k>=2} ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma(2, (3*k)/2 - 1) - PolyGamma(2, (3*(k - 1))/2)))/(48*(k - 1)*(3*k - 4)*(3*k - 2)).
EXAMPLE
-0.0390127263601671660175669413545004617574758571386130990149389673954...
MATHEMATICA
digits = 101;
v[k_] := ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma[2, (3*k)/2 - 1] - PolyGamma[2, (3*(k - 1))/2]))/(48*(k - 1)*(3*k - 4)*(3*k - 2));
V = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]; Join[{0},
RealDigits[V, 10, digits][[1]]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 20 2016
STATUS
approved