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In this note, we prove the existence of a secondary term in the asymptoticformula of the cubic moment of quadratic Dirichlet L-functions$$\sum_{\substack{d - \mathrm{monic \, \& \, sq. \, free} \mathrm{deg}\, d \, =\, D}} L\big(\tfrac{1}{2}, \chi_{d}\big)^{3}$$ over rational function fields onthe order of $q^{\scriptscriptstyle \frac{3}{4} D}.$ This term is in perfectanalogy with the $x^{\scriptscriptstyle \frac{3}{4}}$-term indicated in ourjoint work arXiv:math/0110092v1 for the corresponding asymptotic formula overthe rationals.

On the third moment of $L\big(\frac{1}{2}, \chi_{d}\big)$ I: the rational function field case