  Pazuki and Pengo defined a Northcott property for special values of zeta
functions of number fields and certain motivic $L$-functions. We determine the
values for which the Northcott property holds over function fields with
constant field $\mathbb{F}_q$ outside the critical strip. We then use a case by
case approach for some values inside the critical strip, notably $Re (s) <
\frac{1}{2} - \frac{\log 2}{\log q}$ and for $s$ real such that $1/2 \leq s
\leq 1$, and we obtain a partial result for complex $s$ in the case $1/2<
Re(s)\leq 1$ using recent advances on the Shifted Moments Conjecture over
function fields.
