The classical Eisenstein series are essentially of the form Σ m , n ′ ( ( m + r 1 ) z + n + r 2 ) − s \Sigma ’_{m, n} \left ( (m + r_1) z + n + r_2\right )^{-s} , m m , n n ranging over integer values, Im ⁡ z > 0 \operatorname {Im} z > 0 , r 1 r_1 , r 2 r_2 rational and s s an integer > 2 > 2 . In this paper we show that if s s is taken to be complex the series, with r 1 , r 2 {r_1},{r_2} any real numbers, defines an analytic function of ( z , s ) (z,s) for Im ⁡ z > 0 \operatorname {Im} z > 0 , Re ⁡ s > 2 \operatorname {Re} s > 2 . Furthermore this function has an analytic continuation over the entire s s plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when z z is subjected to a modular transformation is obtained and the special case of s s an integer is studied in detail.