For $k \in \mathbb{N}$ define $$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$ $$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$

By the product for zeta $f(1)=\zeta(2)/\zeta(3)$.

With 100 digits of precision and Cohen's pari script $f(2)=21/(2\pi^2)$ ,$g(1)=315/2\zeta(3)/\pi^4$.

Are the conjectured values correct?

Other closed forms?