Calculation of the Values of the Riemann Zeta Function Via Values of its Derivatives at a Single Point.

We consider the following two problems. We are given the values of several initial derivatives of the Riemann zeta function calculated at some (unknown to us) point a. • How could we calculate an approximate value of the function itself at the same point a without prior finding this number? • How could we find an approximate value of a itself? We suggest several algorithms for answering these questions and demonstrate their accuracy on a few numerical examples. The algorithms reveal some new properties of the zeta function. [ABSTRACT FROM AUTHOR]