The primary focus of this thesis is using dynamical ideas to rapidly compute L-functions. The main results can be summarized as:

Rapid algorithm in the T-aspect. Let Γ be a lattice of SL(2, [special characters omitted]) and let f be a holomorphic or Maass cusp form on Γ\[special characters omitted]. We use the slow divergence of the horocycle flow in Γ\SL(2, [special characters omitted]) to get an algorithm to compute L(f, 1/2+iT) up to a maximum error O( T^−γ) using O( T^7/8+η) operations. Here γ and η are any positive numbers and the constants in O are independent of T. We hence improve the current approximate functional equation based algorithms which have complexity O( T^1+η).

Rapid algorithm in the q-aspect. Let Γ = SL(2, [special characters omitted]), f a modular cusp form on Γ\[special characters omitted] and χ_q be a Dirichlet character on [special characters omitted]/q[special characters omitted]. Let q = MN. Here M = M₁, M₂ such that M₁|N and (M₂, N) = 1, where q, M, N, M₁, M ₂ are integers. We use the dynamics of the Hecke orbits to get an algorithm to compute L(f × χ _q, 1/2) up to any given error O(q ^−γ) using O( M⁵ + N) operations. In the case when q has a factor less than q^1/5, we improve current approximate functional equation based algorithms which need O(q) time complexity. Our algorithm is most effective when q has a suitable factor of size q^1/6.