Rapid computation of L-functions for modular forms

TL;DR

This paper introduces an efficient algorithm for rapidly computing the values of L-functions associated with modular forms at complex points, significantly improving computational speed for large parameters.

Contribution

The paper presents a novel algorithm that computes L-functions for modular forms at high precision with sublinear time complexity in the imaginary part of the input.

Findings

Computes L(f, 1/2 + iT) efficiently for large T

Achieves time complexity of O(1 + |T|^{7/8})

Enables high-precision calculations of L-functions

Abstract

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 $L(f,1/2+ iT)$ to any specified precision in time $O(1+|T|^{7/8})$.