[math-fun] number theory question

since 1/Product[(1 - 1/Prime[k]^w), {k, \[Infinity]}] equals Zeta[w], I wondered what this product becomes when specialising for either primes p with 4|(p+1) or primes q with 4|(q-1). Numerics at w=2 point to 1.690 and 1.335 aboutish. W.