(OEIS A175639), where the product is over the primes .
Taking the logarithm, expand the sum about infinity, and then summing the terms gives
a "closed" form as

(3)

(4)

where
is the prime zeta function and the s are rational numbers given as the coefficients of in the series

Finch, S. "Class Number Theory." http://algo.inria.fr/csolve/clss.pdf. May 6, 2005.Sloane, N. J. A. Sequence A175639
in "The On-Line Encyclopedia of Integer Sequences."Taniguchi,
T. "A Mean Value Theorem for the Square of Class Number Times Regulator of Quadratic
Extensions." http://arxiv.org/abs/math/0410531.
3 Jul 2006.