Reimann introduced the study of the zeta function of a complex variable, and showed how multiplying zeta by a gamma factor and other simple factors resulted in an even, entire function called the Riemann Xi-function. He conjectured that all the zeros of the Xi-function are real, now known as the Riemann Hypothesis. In this talk we introduce the study of the zeros of the partial sums in Riemann's uniformly convergent series expansion for the Xi function in terms of incomplete gamma functions, and discuss how various known or conjectured properties of the Xi-function seem to be reflected by these approximates.