  We establish a uniform approximation result for the Taylor polynomials of the
xi function of Riemann which is valid in the entire complex plane as the degree
grows. In particular, we identify a domain growing with the degree of the
polynomials on which they converge to Riemann's xi function. Using this
approximation we obtain an estimate of the number of "spurious zeros" of the
Taylor polynomial which are outside of the critical strip, which leads to a
Riemann - von Mangoldt type of formula for the number of zeros of the Taylor
polynomials within the critical strip. Super-exponential convergence of Hurwitz
zeros of the Taylor polynomials to bounded zeros of the xi function are
established along the way, and finally we explain how our approximation
techniques can be extended to a collection of analytic L-functions.
