$a$-Points of the Riemann zeta-function on the critical line

We investigate the proportion of the nontrivial roots of the equation $\zeta (s)=a$, which lie on the line $\Re s=1/2$ for $a \in \mathbb C$ not equal to zero. We show that at most one-half of these points lie on the line $\Re s=1/2$. Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to $\zeta (s)=a$ lie on the line $\Re s=1/2$ for any nonzero complex number $a$.