Author’s summary: Let \(E\) be an elliptic curve defined over \(\mathbb Q\) and let \(K/\mathbb Q\) be a finite Galois extension with Galois group \(G\). The equivariant Birch–Swinnerton-Dyer conjecture for \(h^1(E\times_{\mathbb Q}K)(1)\) viewed as a motive over \(\mathbb Q\) with coefficients in \(\mathbb Q[G]\) relates the twisted \(L\)-values associated with \(E\) with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an \(S_3\)-extension of \(\mathbb Q\).