Dynamical methods for rapid computations of L-functions

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 
$L(f, 1/2 +iT)$ upto 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 $O(T^{1+\eta})$.