  In this paper we study iterated Eisenstein {\tau}-integrals and multiple
Eisenstein L-series, they are functions on the complex upper half plane and
form two Q-algebras. They reduce to iterated Eisenstein integrals and multiple
Hecke L-functions with respect to Eisenstein series respectively after analytic
extension when {\tau}->0. We give the relations among them and prove the linear
independence of their elements. Finally, we explain the connections among
double Eisenstein L-functions and holomorphic double modular values.
