An accurate approximation of zeta-generalized-Euler-constant functions

Zeta-generalized-Euler-constant functions, gamma(s) := Sigma(infinity)(k=1) (1/k(s) - integral(k+1)(k) dx/x(s)) and (gamma) over tilde (s) := Sigma(infinity)(k=1)(-1)(k+1) (1/k(s) - integral(k+1)(k) dx/x(s)), defined on the closed interval [0, infinity), where gamma(1) is the Euler-Mascheroni constant and (gamma) over tilde (1) = ln 4/pi, are studied and estimated with high accuracy.