Apery proved in 1976 that $\zeta(3)$ is irrational, and we know that for all integers $n$, \[\zeta(2n)=\alpha \pi^{2n}\] for some $\alpha\in \mathbb{Q}$. Given these facts, it seems natural to ask whether we can have \[ \zeta(n)=\alpha \pi^n\]

for all $n$ (I'm mainly interested in the case $n=3$). The proofs I've seen of the irrationality of $\zeta(3)$ don't seem to give this information.

My gut feeling is that the answer is no, but I can't find any reference proving this fact. I know that the answer hasn't been proven to be yes ($\zeta(3)$ isn't even known to be transcendental), but ruling out this possibility seems an easier problem.