Is it known if a functional equation of the form:
$$\zeta (s)=f(s) \zeta (s+1)$$
can exist?
If it is possible for such a functional equation to exist then I believe lots of wonderful things would happen. In particular one could solve this integral:
$$\int \log (\zeta (s)) \, ds$$
on the critical line by extending the validity of the Euler product formula to it.