In view of the relationship with the Krätzel function, we derive a new series representation for the λ-generalized Hurwitz-Lerch Zeta function introduced by H.M. Srivastava [Appl. Math. Inf. Sci. 8 (2014) 1485–1500] and determine the monotonicity of its coefficients. An integral representation of the Mathieu (a, λ)-series is rederived by applying the Abel's summation formula (which provides a slight modification of the result given by Pogány [Integral Transforms Spec. Funct. 16 (8) (2005) 685–689]) and this modified form of the result is then used to obtain a new integral representation for Srivastava's λ-generalized Hurwitz-Lerch Zeta function. Finally, by making use of the various results presented in this paper, we establish two sets of two-sided inequalities for Srivastava's λ-generalized Hurwitz-Lerch Zeta function.