Computation of classical and v-adic L-series of t-motives

We design an algorithm for computing the L-series associated to an Anderson t-motive, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that, after renormalization by the local factor at v, the order of vanishing at T=1 of the v-adic L-series of a given Anderson t-motive does not depend on the finite place v.