Motivated by the work of E. Witten on the symplectic volume of the space of homomorphisms of the fundamental group of a Riemann surface, D. Zagier introduced a series and gave a residue formula for it, by considering a sequence of linear forms in complex variables with integral coefficients for some numbers related with Bernoulli numbers. The authors of this paper define the Eisenstein series by introducing the oscillating factor and give an explicit residue formula for it. By using the Eisenstein series, another proof of the Szenes residue formula is given.