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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 ofthe 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 thegroup of unitary matrices multiplied by a weight function.

Expected Values of $L$-functions Away from the Central Point