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A306016 Decimal expansion of -Zeta'(2)/Zeta(2).

%I A306016 #24 Jun 13 2021 13:27:04
%S A306016 5,6,9,9,6,0,9,9,3,0,9,4,5,3,2,8,0,6,3,9,9,8,6,4,3,6,0,0,1,9,7,3,0,0,
%T A306016 0,2,4,0,3,4,8,2,2,8,0,8,0,6,9,3,0,9,7,9,5,5,8,1,9,7,3,6,0,4,3,7,9,1,
%U A306016 7,2,7,7,3,6,6,7,4,0,6,0,6,7,8,7,8,6,7
%N A306016 Decimal expansion of -Zeta'(2)/Zeta(2).
%H A306016 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 179.
%H A306016 J. B. Rosser, L. Schoenfeld, <a href="https://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Ill. J. Math. 6 (1) (1962) 64-94, Table IV.
%F A306016 Equals -(6/Pi^2)*Zeta'(2).
%F A306016 Equals 1 - 12*Zeta'(-1) - gamma - log(2*Pi).
%F A306016 From _Amiram Eldar_, Aug 14 2020: (Start)
%F A306016 Equals Sum_{k>=1} Lambda(k)/k^2, where Lambda is the Mangoldt function.
%F A306016 Equals Sum_{p prime} log(p)/(p^2 - 1). (End)
%e A306016 Equals 0.569960993094532806399864360019730002403482280806930979558...
%p A306016 -(6/Pi^2)*Zeta(1,2): evalf(%,100);
%t A306016 RealDigits[- Zeta'[2] / Zeta[2], 10, 87][[1]]
%o A306016 (PARI) -zeta'(2)/zeta(2) \\ _Michel Marcus_, Jan 11 2019
%Y A306016 Cf. A059956, A073002.
%K A306016 nonn,cons
%O A306016 0,1
%A A306016 _Peter Luschny_, Jun 17 2018