OFFSET
0,1
LINKS
Ramanujan's Papers, Some formulas in the analytic theory of numbers Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).
FORMULA
D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279.
EXAMPLE
0.4603233722587214303937620863844189747632149035387392240584250348445902629324...
MATHEMATICA
24*EulerGamma^3/Pi^2 - (432*Zeta'[2] /Pi^4+ 36/Pi^2)*EulerGamma^2 + (3456*Zeta'[2]^2/Pi^6 + 288*(Zeta'[2]-Zeta''[2])/Pi^4 + 24/Pi^2 - 72*StieltjesGamma[1]/Pi^2)*EulerGamma + StieltjesGamma[1]*(288*Zeta'[2]/Pi^4 + 24/Pi^2)-10368*Zeta'[2]^3/Pi^8 - 864*Zeta'[2]^2/Pi^6 + 1728*Zeta''[2] * Zeta'[2]/Pi^6 + 72*(Zeta''[2]-Zeta'[2])/Pi^4 - 48*Zeta'''[2]/Pi^4 + (12*StieltjesGamma[2] - 6)/Pi^2
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Sep 10 2018
STATUS
approved