Rapid computation of L-functions for modular forms

Let $f$ be a fixed (holomorphic or Maass) modular cusp form, with $L$-function $L(f,s)$. We describe an algorithm that computes the value $L(f,1\mathrm{/}2+iT)$ to any specified precision in time $O(1+\mathrm{\mid }T{\mathrm{\mid }}^{7\mathrm{/}8})$.