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^∨,   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.