Twenty Digits of Some Integrals of the Prime Zeta Function

The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645… over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen’s work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.