Special values of Goss L-series attached to Drinfeld modules of rank 2

Inspired by the classical setting, Goss defined the $L$-series of Drinfeld $A$-modules corresponding to representations of the absolute Galois group of a rational function field. In this paper, for a given Drinfeld $A$-module $\phi$ of rank 2 defined over the finite field $\mathbb{F}_q$, we give explicit formulas for the values of Goss $L$-series at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.