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We prove the asymptotic formulae for several moments of derivatives of GL(2)  L -functions over quadratic twists. The family of  L -functions we consider has root number fixed to −1 and odd orthogonal symmetry. Assuming generalized Riemann hypothesis, we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms  f  and  g , and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves using the modularity theorem and the work of Gross and Zagier.

Moments of L ′(½) in the Family of Quadratic Twists