  Inspired by the classical setting, Goss defined $L$-series attached to
Drinfeld modules. In this paper, for a fixed choice of a power $q$ of a prime
number and a given Drinfeld module $\phi$ of rank 2 with a certain condition on
its coefficients, we give explicit formulas for the values of Goss $L$-series
attached to $\phi$ at positive integers $n$ such that $2n+1\leq q$ in terms of
polylogarithms and coefficients of the logarithm series of $\phi$.
