  We design an algorithm for computing the $L$-series associated to an Anderson
$t$-motives, exhibiting quasilinear complexity with respect to the target
precision. 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 good
reduction does not depend on the finite place $v$.
