A couple of years ago, I asked this MSE question on the evaluation of the product of even zeta values: $$ \prod_{n=1}^\infty \zeta(2n) \approx 1.82 \quad .$$ While it can be shown that the product converges, the exact value is -- as far as I know -- unknown (please correct me if this isn't the case).
I wonder to what extent more general products of (Riemann) zeta values, e.g.
$$\prod_{n=1}^{\infty} a_{n} \zeta(c_{n} \cdot n), $$ for some sequences $(a_{n})$ and $(c_{n})$ have been studied, and what kind of results have been obtained. Even though a thing or two is known about rational zeta series, information on their product counterparts seems to be sparse. Any pointers to relevant literature are much appreciated.
