I was wondering if a zero on the critical line implies no zero for the zeta function anywhere else in the critical strip for the same ordinate and vice-versa? I don't know if there is a proof for this.
That is does,
$\zeta(\frac{1}{2} + bi) = 0 \implies \zeta(a + bi) \neq 0$ for $0 < a <\frac{1}{2}$ and $\frac{1}{2} < a < 1$
Thank you