  The aim of the present note is to develop a study on the feasibility of a
unified theory of mean values of automorphic L-functions, a desideratum in the
field. This is an outcome of the investigation commenced with Part XII of this
series, where a framework was laid on the basis of the theory of automorphic
representations, and a general approach to the mean values was envisaged.
Specifically, it is shown here that the inner-product method, which was
initiated by A. Good and greatly enhanced by M. Jutila, ought to be brought to
perfection so that the mean square of the L-function attached to any cusp form
on the upper half-plane could be reached within the notion of automorphy. The
Kirillov map is our key implement. Because of its geometric nature, our
argument appears to extend to bigger linear Lie groups. This paper is
essentially self-contained.
