  In this paper various analytic techniques are com- bined in order to study
the average of a product of a Hecke L- function and a symmetric square
L-function at the central point in the weight aspect. The evaluation of the
second main term relies on the theory of Maa{\ss} forms of half-integral weight
and the Rankin-Selberg method. The error terms are bounded using the
Liouville-Green approximation.
