We study the low-lying zeros of L-functions attached to quadratic twists of a
given elliptic curve E defined over Q. We are primarily interested in
the family of all twists coprime to the conductor of E and compute a very
precise expression for the corresponding 1-level density. In particular, for
test functions whose Fourier transforms have sufficiently restricted support,
we are able to compute the 1-level density up to an error term that is
significantly sharper than the square-root error term predicted by the
L-functions Ratios Conjecture.Comment: 33 page