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

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