Special L-values of t-motives: a conjecture

We propose a conjecture on special values of $L$-functions in a function field context with positive characteristic coefficients. For $M$ a uniformizable $t$-motive with everywhere good reduction we conjecture a relation between the value of the Goss $L$-function $L(M^\vee, s)$ at $s = 0$ and the uniformization of the abelian $t$-module associated with $M$. When $M$ is a power of the Carlitz $t$-motive the conjecture specializes to a theorem of Anderson and Thakur on Carlitz zeta values. Beyond this case we present numerical evidence.