Let \(E\) be an elliptic curve defined over \(\mathbb Q\), and assume that \(E\) has complex multiplication by the ring of a complex quadratic number field \(K\) with class number \(1\). Y. Tian [Proc. Natl. Acad. Sci. USA 109, No. 52, 21256–21258 (2012; Zbl 1298.11053); Camb. J. Math. 2, No. 1, 117–161 (2014; Zbl 1303.11067)] discovered how to derive results about the arithmetic of certain quadratic twists of \(E\) with root number \(-1\) from a weak form of the \(2\)-part of the conjecture of Birch and Swinnerton-Dyer for quadratic twists of \(E\) with root number \(+1\). Zhao managed to prove such a weak form of BSD in a series of recent articles [C. Zhao, Math. Proc. Camb. Philos. Soc. 121, No. 3, 385–400 (1997; Zbl 0882.11039); Math. Proc. Camb. Philos. Soc. 121, No. 3, 385–400 (1997; Zbl 0882.11039); Math. Proc. Camb. Philos. Soc. 131, No. 3, 385–404 (2001; Zbl 1042.11038); Math. Proc. Camb. Philos. Soc. 134, No. 3, 407–420 (2003; Zbl 1053.11053)], and the aim of the article under review is showing that Zhao’s elementary method becomes even simpler when applied to quadratic twists of elliptic curves with good reduction at \(2\). In more technical terms, the authors prove a lower bound for the \(2\)-adic valuation of the algebraic part of the value of the \(L\)-series of a family of quadratic twists of \(E\) at \(s = 1\), and this lower bound agrees with the prediction by the conjecture of Birch and Swinnerton-Dyer.