  Recently Katz and Sarnak introduced the idea of a symmetry group attached to
a family of L-functions, and they gave strong evidence that the symmetry group
governs many properties of the distribution of zeros of the L-functions. We
consider the mean-values of the L-functions and the mollified mean-square of
the L-functions and find evidence that these are also governed by the symmetry
group. We use recent work of Keating and Snaith to give a complete description
of these mean values. We find a connection to the Barnes-Vign\'eras
$\Gamma_2$-function and to a family of self-similar functions.
