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 nontrivial zeros of $\zeta$ have irrational imaginary part. Has this question been studied? Any reference?
Thank you in advance.
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This set includes the nonpositive integers (including the trivial zeros) so the set is infinite. I would be interested in another example. 


See D. Maser's Rational Values of the Riemann zeta function. In it he proves that the number of rational numbers $s\in[2,3]$ with denominator at most $D$ such that $\zeta(s)$ is also a rational with denominator at most $D$ is $O\left( \left(\frac{\log(D)}{\log\log(D)}\right)^2 \right)$. Of course the set is probably empty. 

