Towards non-iterative calculation of the zeros of the Riemann zeta function

We introduce a family of rational functions RN(a,d0,d1,...,dN) with the following property. Let d0,d1,...,dN be equal respectively to the value of some function f(s) and the values of its first N derivatives calculated at a certain complex number a lying not too far from a zero rho of this function. It is expected that the value of RN(a,d0,d1,...,dN) is very close to rho We demonstrate this phenomenon on several numerical instances with the Riemann zeta function in the role of f(s). For example, for N=10 and a=0.6+14i we have |R-10(a,d(0),d(1),...,d(10))-rho 1|<10-18 where rho 1=0.5+14.13...i is the first non-trivial zeta zeroAlso we define rational functions R-N,n(a,d(0),d(1),...,d(N)) which (under the same assumptions) have values which are very close to n-rho, that is, to the terms from the Dirich let series for the zeta function calculated at its zeroIn the case when a is between two consecutive zeros, say rho l and rho l+1,, functions R-N,n(a,d(0),d(1),...,d(N)) approximate neither of n-rho lnon-rho l+1 ; nevertheless, they allow us to approximate the sum n-rho l+n-rho l+1 and the product n-rho l+n-rho l+1 and hence to calculate both n-rho ln-rho l+1 by solving corresponding quadratic equation.