A well known form of Euler product is
$ \prod_p^\infty \big(1- \frac{1}{ p^{s} } \big) = \frac{1}{ \zeta (s) } $
for all primes $p$.
Are there any other variations of above product, such as :
$ \prod_p^\infty \big(1- \frac{k}{ p^{s} } \big)$ for any integer $ k > 1$
or
$ \prod_p^\infty \big(1- \frac{f (p)}{ p^{s} } \big)$ where $f(p)$ is any non zero polynomial of prime $p$.
Few of them are discussed on the Wikipedia page for Euler product
But the page only includes numerical values of such products, and none of them are represented in terms of other well known constants like $\pi$ or in terms of other Zeta functions like $\zeta(s)$.
Kindly let me about any such publication which discussed such variation in Euler product in detail.
