Absolute values of L-functions for GL(n,R) at the point 1

Abstract We study the values of | L ( 1 , F ) | for Hecke–Maass cusp forms F on S L ( n , Z ) ( n ≥ 3 ) of large Langlands parameters. New unconditional results on the extreme values and conditional results on the size range are derived, which determine precisely the order of magnitude of L ( 1 , F ) . In addition, we enhance the new average estimate toward the Ramanujan Conjecture due to Matz and Templier. An application of the Hecke multiplicativity to the Littlewood–Richardson rule for a product of two Schur polynomials is cultivated.