Starting from the known summation formula for the Riemann zeta function of odd positive integer order, we generalize the factorial to the Gamma function and establish a fast convergent summation formula suitable for real order s > 1. We first give an integral representation of the zeta function of real order and rigorously prove its equivalence to the series representation. This formula unifies the cases of odd and even orders, revealing the essential connection between the zeta function, the circle constant π,and Bernoulli numbers. We provide complete mathematical derivations, including convergence analysis and special value verification. Numerical experiments show that the formula exhibits super-exponential convergence; for s > 2, typically only the first 10-20 terms are needed to achieve double-precision accuracy.Furthermore, we prove that this formula also holds in the region 0 < s ≤ 1, thus providing a new perspective on analytic continuation. We discuss its potential value in computation and applications,and present detailed comparisons with existing algorithms such as the Riemann-Siegel formula.
shifa liu (Wed,) studied this question.