Let $f$ be a holomorphic or Maass cusp form on the upper half plane. We use the slow divergence of the horocycle flow on the upper half plane to get an algorithm to compute $L(f,1/2+iT)$ up to a maximum error $O(T^{-\gamma})$ using $O(T^{7/8+\eta})$ operations. Here $\gamma$ and $\eta$ are any positive numbers and the constants in $O$ are independent of $T$. We thus improve the current approximate functional equation based algorithms which have complexity $O(T^{1+\eta})$.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/246