Explicit bounds for L-functions on the edge of the critical strip

Assuming GRH and the Ramanujan-Petersson conjecture we prove explicit bounds for L(1, f) for a large class of L-functions L(s, f), which includes L-functions attached to automorphic cuspidal forms on GL(n). The proof generalizes work of Lamzouri, Li and Soundararajan. Furthermore, the main results improve the classical bounds of Littlewood (1 + o(1)) (12e(r)/pi(2) log log C(f)) (-d) <= vertical bar L(1, f)vertical bar <= (1 + o(1)) (2e(r) log log C (f)) (d), where C(f) is the analytic conductor of L(s, f). (C) 2018 Elsevier Inc. All rights reserved.