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In this paper, we prove the extreme values of $L$-functions at the centralpoint for almost prime quadratic twists of an elliptic curve. As anapplication, we get the extreme values for the Tate--Shafarevich groups in thequadratic twist family of an elliptic curve under the Birth--Swinnerton-Dyerconjecture.

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