Provable computation of motivic l-functions

L-functions have been a central object of study in number theory ever since the discovery of the Riemann zeta function, and are still an area of active research. The behavior of the L-function at specific points called special values often gives global information about the object to which it is attached. We give an algorithm to provably compute values and derivatives of L-functions at arbitrary points on the complex plane using their functional equations, and give several applications of this algorithm, in particular computing Heegner points and investigations of the Birch and Swinnerton-Dyer conjecture.