  Andrade and Keating computed the mean value of quadratic Dirichlet
$L$--functions at the critical point, in the hyperelliptic ensemble over a
fixed finite field $\mathbb{F}_q$. Summing $L(1/2,\chi_D)$ over monic,
square-free polynomials $D$ of degree $2g+1$, the main term is of size $|D|
\log_q |D|$ (where $|D|=q^{2g+1}$) and Andrade and Keating bound the error term
by $|D|^{\frac 34+ \frac{\log_q(2)}{2}}$. For simplicity, we assume that $q$ is
prime with $q \equiv 1 \pmod 4$. We prove that there is an extra term of size
$|D|^{1/3} \log_q|D|$ in the asymptotic formula and bound the error term by
$|D|^{1/4+\epsilon}$.
