  In this paper we use techniques first introduced by Florea to improve the
asymptotic formula for the first moment of the quadratic Dirichlet L-functions
over the rational function field, running over all monic, square-free
polynomials of even degree at the central point. With some extra technical
difficulties that doesn't appear in Florea's paper, we prove that there are
extra main terms of size $(2g+2)q^{\frac{2g+2}{3}},
q^{\frac{g}{6}+\left[\frac{g}{2}\right]}$ and
$q^{\frac{g}{6}+\left[\frac{g-1}{2}\right]}$, whilst bounding the error term by
$q^{\frac{g}{2}(1+\epsilon)}$.
