Approximation of Riemann's zeta function by finite Dirichlet series:   multiprecision numerical approach
  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. 
Gleb Beliakov and Yuri Matiyasevich
math.NT