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In this paper we use techniques first introduced by Florea to improve theasymptotic formula for the first moment of the quadratic Dirichlet L-functionsover the rational function field, running over all monic, square-freepolynomials of even degree at the central point. With some extra technicaldifficulties that doesn't appear in Florea's paper, we prove that there areextra 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$q^{\frac{g}{2}(1+\epsilon)}$.

The First Moment of $L(\frac{1}{2},\chi)$ for Real Quadratic Function Fields