Extreme central L-values of almost prime quadratic twists of elliptic curves

In this paper,we prove the extreme values of L-functions at the central point for almost prime quadratic twists of an elliptic curve.As an application,we get the extreme values for the Tate-Shafarevich groups in the quadratic twist family of an elliptic curve under the Birch and Swinnerton-Dyer conjecture.