An Analytical and Numerical Detour for the Riemann Hypothesis

From the functional equation F(s)=F(1-s) of Riemann's zeta function, this article gives new insight into Hadamard's product formula. The function S-1(s)=d(lnF(s))/ds and its family of associated S-m functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the F function. This family is a mathematical and numerical tool which makes it possible to estimate the value F(s) of the function at a point s=x+iy=x+1/2+iy in the critical strip S from a point ???=1/2+iy on the critical line Script capital L.Generating estimates S-m*(s) of S-m(s) at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstrass canonical form. A final and brief comparison is made with the zeta and F functions over finite fields.