Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189

The Generalized Riemann Hypothesis (GRH) states that all nontrivial zeros of Dirichlet L-functions lie on the line Re(s) = 1 2 .Further, it is believed that there are no Q-linear relations among the nonnegative ordinates of these zeros.In particular, it is expected that L( 1 2 , χ) = 0 for all primitive characters χ, but this remains unproved.It appears to have been conjectured first by S. D. Chowla 5 in the case when χ is a quadratic character.In addition to numerical evidence (see 16 and 17) the philosophy of N.Katz and P. Sarnak 13 lends theoretical support to this belief.Assuming the GRH, they proved that (oral communication) for at least (19Independently, A. E. Özluk and C. Snyder 15 showed, also assuming GRH, that L( 1 2 , χ d ) = 0 for at least 15 16 of the fundamental discriminants |d| ≤ X.Katz and Sarnak also developed conjectures on the low-lying zeros in this family of L-functions (analogous to the Pair Correlation conjecture regarding the vertical distribution of zeros of ζ(s)) which imply that L( 1 2 , d • ) = 0 for almost all fundamental discriminants d.In a different vein, R. Balasubramanian and V. K. Murty 1 showed that for a (small) positive proportion of the characters (mod q), L( 1 2 , χ) = 0. Recently, H. Iwaniec and P. Sarnak 10 have demonstrated that this proportion is at least one third.For integers d ≡ 0, or 1 (mod 4) we put χ d (n) = d n .Notice that χ d is a real character with conductor ≤ |d|.If d is an odd, positive, square-free integer then χ 8d is a real, primitive character with conductor 8d, and with χ 8d (-1) = 1.In 19, we considered the family of quadratic twists of a fixed Dirichlet L-function L(s, ψ).Precisely, we considered the family L(s, ψ ⊗ χ 8d ) for odd, positive, square-free integers d.When ψ is not quadratic we showed that at least 1 5 of these L-functions are not zero at s = 1 2 , and indicated how this proportion may be improved to 1 3 .The most interesting case when