  The finite Dirichlet series from the title are defined by the condition that
they vanish at as many initial zeroes of the zeta function as possible. It
turned out that such series can produce extremely good approximations to the
values of Riemann's zeta function inside the critical strip. In addition, the
coefficients of these series have remarkable number-theoretical properties
discovered in large scale high accuracy numerical experiments.
  So far no theoretical explanation to the observed phenomena was found.
