It is well known that if $\rho$ is a nontrivial zero of $\zeta(s)$ then so are $1-\rho$, $\bar{\rho}$, and $1-\bar{\rho}$. My question is: do these four zeros necessarily have the same multiplicity?
I know that it is unknown, but widely believed, that all zeros of $\zeta$ are simple, but given a nontrivial zero $\rho$, with multiplicity $n$, are the multiplicities of $1-\rho$, $\bar{\rho}$, and $1-\bar{\rho}$ also $n$?
I read an article stating that it has been shown that $\zeta'(s)\neq0$ when $\Re(s)<1/2$, which would immediately answer my question with a "no," but I also saw in a different article that $\zeta'$ having any zeros on the left side of the critical line $s=1/2$ would imply the Riemann Hypothesis, so if that's the case then it is still unverified and does not answer my question.