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Decimal expansion of -Zeta'(2)/Zeta(2).
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%I #24 Jun 13 2021 13:27:04

%S 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 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 7,2,7,7,3,6,6,7,4,0,6,0,6,7,8,7,8,6,7

%N Decimal expansion of -Zeta'(2)/Zeta(2).

%H 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 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 Equals -(6/Pi^2)*Zeta'(2).

%F Equals 1 - 12*Zeta'(-1) - gamma - log(2*Pi).

%F From _Amiram Eldar_, Aug 14 2020: (Start)

%F Equals Sum_{k>=1} Lambda(k)/k^2, where Lambda is the Mangoldt function.

%F Equals Sum_{p prime} log(p)/(p^2 - 1). (End)

%e Equals 0.569960993094532806399864360019730002403482280806930979558...

%p -(6/Pi^2)*Zeta(1,2): evalf(%,100);

%t RealDigits[- Zeta'[2] / Zeta[2], 10, 87][[1]]

%o (PARI) -zeta'(2)/zeta(2) \\ _Michel Marcus_, Jan 11 2019

%Y Cf. A059956, A073002.

%K nonn,cons

%O 0,1

%A _Peter Luschny_, Jun 17 2018