Is $p^s$ transcendental if $\zeta(s)=0$?

Question

Let $K$ be a number field and $\mathcal{O}_{K}$ its ring of integers.

Let

$$\zeta_{K}(s)=\prod_{\mathfrak{m}}\frac{1}{1-\#(\mathcal{O}_{K}/\mathfrak{m})^{-s}}$$ be the $\zeta$ function associated to $K$ (more precisely: the analytic continuation thereof to the complex plane).

Let $p$ be a prime and $s$ a zero of $\zeta_{K}$ with $\operatorname{Re}(s)>0$.

Question: Is $p^s$ transcendental?

This would of course be true $s$ were algebraic, as a consequence if the Gelfond-Schneider theorem.

I believe that all such zeros are at least conjectured to be transcendental.