  In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of
families of GL_2 automorphic L-functions was investigated. In this paper we
examine the distribution of zeros which are at or neat s=1/2 (that is the
central point) for such families of L-functions. Unlike [IS], most of the
results in this paper are conditional, depending on the Generalized Riemann
Hypothesis (GRH). It is by no means obvious, but on the other hand not
surprising, that this allows us to obtain sharper results on nonvanishing.
