Let \(F\) be a non-Archimedean local field of characteristic zero. Let \(G\) be the \(F\)-points of a reductive algebraic group over \(F\) and let \(H\) be the \(F\)-points of a reductive subgroup of \(G\) over \(F\). An irreducible representation, say \(\pi\), of \(G\) is said to be \(\chi\)-distinguished with respect to \(H\), where \(\chi:H\to C^x\) is a character of \(H\), if it admits a non-trivial \((H,\chi)\)-equivariant linear form, which is to say \(\operatorname{Hom} H(\pi,\chi)\neq 0\). Let \(E/F\) be a quadratic extension of non-Archimedean local fields of characteristic zero. An irreducible admissible representation \(\pi\) of \(\mathrm{GL}(n,F)\) is said to be distinguished with respect to \(\mathrm{GL}(n,F)\) if it admits a non-trivial linear form that is invariant under the action of \(\mathrm{GL}(n,F)\). It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the \(F\)-points of the mirabolic subgroup when \(\pi\) is unitary and generic. The authors of the paper under review prove that the essential vector of Jacquet, Piatetski-Shapiro and Shalika is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local \(L\)-value. They extend all their results to the non-unitary generic case. The paper is very well written with very deep results.