  Fixing $t \in \mathbb{R}$ and a finite field $\mathbb{F}_q$ of odd
characteristic, we give an explicit upper bound on the proportion of genus $g$
hyperelliptic curves over $\mathbb{F}_q$ whose zeta function vanishes at
$\frac{1}{2} + it$. Our upper bound is independent of $g$ and tends to $0$ as
$q$ grows.
