Averages of coefficients of the decomposable <i>L</i>-function of degree seven

Let [Formula: see text] be a Hecke–Maass cusp form for [Formula: see text] with normalized Hecke eigenvalues [Formula: see text] and f be a Hecke–Maass cusp form for [Formula: see text] with normalized Hecke eigenvalues [Formula: see text]. Under the generalized Ramanujan conjecture, we give a detailed proof of an asymptotic formula for averages of coefficients of degree seven L-functions which can be factorized as a product of degree one and degree six L-functions, namely [Formula: see text] We improve the results of Friedlander and Iwaniec [ 2 ] by changing the error term from [Formula: see text] to [Formula: see text]. Our proof is based on methods from the bilinear structure in the dual sum.