Étude de $L(s, \chi)/\pi^s$ pour des fonctions $L$ relatives à $\mathbb{F}_q((T^{-1}))$ et associées à des caractères de degré $1$

For functions fields, Carlitz defined an analogue of the real π and Goss defined the analogue of Dirichlet L-functions. We prove in a particular case that there exist integer values s and characters χ such that L(s,χ)/π 8 is rational, algebraic or transcendent.