[Part III, cf. Théorie des nombres, Sémin., Paris/Fr. 1989-90, Prog. Math. 102, 113-120 (1992; Zbl 0763.11024)] The present paper is a continuation of the authors' previous work on estimates for the coefficients of ordinary Dirichlet series satisfying a standard functional equation. The main result (roughly speaking) is the following: Let \(A(s)\) be an ordinary Dirichlet series that converges absolutely for \(\text{Re} (s)>1\) and when multiplied with \(k\) gamma factors of the usual type has a functional equation under \(s\mapsto 1-s\). Assume furthermore that the same is true for the twisted \(L\)-series \(A(s, \chi)\) where \(\chi\) runs through almost all even characters of prime moduli \(p\) in a set of positive density. Then the Rankin series \(A_ 2 (s)\) obtained by squaring the coefficients of \(A(s)\) converges absolutely for \(\text{Re} (s)> 2{{k-1} \over k}\). As an application the authors deduce the bound \(| \lambda_ n |\leq n^{15/76} \sigma_ 0 (n)\) for the eigenvalues \(\lambda_ n\) of a Maass cusp form on \(\text{SL}_ 2 (\mathbb{Z})\). The latter is sharper than the bound \(| \lambda_ p |< p^{1/5}+ p^{-1/5}\) (\(p\) prime) obtained before by \textit{F. Shahidi} [Automorphic forms and analytic number theory, Proc. Conf., Montreal/Canada 1989, 135-141 (1990; Zbl 0748.11025)], but as the authors' remark, meanwhile has been surpassed by work of \textit{D. Bump}, \textit{J. Hoffstein} and the authors [Int. Math. Res. Not. 1992; No. 4, 75-81 (1992; Zbl 0760.11017)].