  In this paper we investigate some consequences of the Gross/Zagier types of
formulae which were introduced by Gross and Zagier, and were then generalized
in various directions by Hatcher, Zhang, Kudla and several other people.
Working in the classical context of central values of L-series of holomorphic
forms of prime level, we deduce an exact average formula for suitable twists of
such L-values, with a relation to the class number of associated imaginary
quadratic fieds, thereby strengthening a result of Duke. One also obtains a
stability result, as well as subconvexity (in this setting), and certian
non-vanishing assertions. This article is dedicated to the memory of Serge
Lang.
