Secondary terms in the number of vanishings of quadratic twists of   elliptic curve L-functions
  We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions. 
J. Brian Conrey, Atul Pokharel, Michael O. Rubinstein, Mark Watkins