  We compute the expected value of Dirichlet $L$-functions defined over
$\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in
(0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of
the transition at the point $s=\frac{1}{2}$ of the bound for the size of an
$L$-function implied by the Lindel\"of hypothesis. We show that at
$s=\frac{1}{3}$, the expected value matches corresponding statistics of the
group of unitary matrices multiplied by a weight function.
