In the paper, a generalization of the asymptotic expansions obtained by \textit{M.~Katsurada} [Proc.~Japan Acad. 74, No. 10, 167--170 (1998; Zbl 0937.11035)] and \textit{D.~Klusch} [J.~Math. Anal. Appl. 170, No. 2, 513--523 (1992; Zbl 0763.11036)] for the Lipschitz-Lerch zeta function \[ R(a, x, s)\equiv\sum_{k=0}^\infty {e^{2k\pi ix}\over (a+k)^s}, \quad s, x, a \in \mathbb C, \quad 1-a\notin \mathbb N, \quad \Im x \geq 0, \] to the Hurwitz-Lerch zeta function \[ \Phi(z, s, a)\equiv\sum_{k=0}^\infty {z^k\over (a+k)^s}, \quad 1-a\notin \mathbb N, \quad | z| 0, \quad \Re s>0, \quad z\notin[1, \infty), \] given in [\textit{H.~M.~Srivastava} and \textit{J.~Choi}, Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1014.33001)], the authors obtain an integral representation which gives the analytical continuation of the function \(\Phi(z, s, a)\) to the region \(z\in\mathbb C\setminus[1, \infty)\) if \(\Re a>0\), and \(z\in\{z\in \mathbb C, | z| <1\}\) if \(\Re a \leq 0\), \(a\in \mathbb C\setminus\mathbb R^-\). From this they deduce three complete asymptotic expansions for either large or small \(a\) and large \(z\) with error bounds. Moreover, the numerical examples for these bounds are presented.