Random Matrix Theory and L - Functions at s = 1 / 2

Recent results of Katz and Sarnak [8,9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s = 1/2 and those of the characteristic polynomials Z(U, θ) of matrices U with respect to averages overSO(2N) andUSp(2N) at the corresponding point θ = 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments ofZ(U, 0) for each ensemble, and hence calculate the asymptotic (large N ) value distributions forZ(U, 0) and logZ(U, 0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point.