The question is the title. For example, if we could show that $S$ is finite, then this would entail that every large enough integer $n$ is such that $\zeta(2n+1)$ is irrational and that, under RH, almost all non-trivial zeros of $\zeta$ have irrational imaginary part. Has this question been studied? Any reference?  

Thank you in advance.



What is known about the set $S$ of couples of rationals $(q,q')$ such that $\zeta(q+iq')$ is rational?
This set includes the non-positive integers (including the trivial zeros)
so the set is infinite.


I would be interested in another example.


