The currently available methods of finding the zeros of a vector function are quite effective but they do require a prior knowledge of the approximate locations of the zeros. Such a knowledge is, ...

“We measure the partition function by determining where it is zero. Once you know all the zeros of a function, you know the whole function.

Marcus du Sautoy, Fritz Grunewald, Zeta Functions of Groups: Zeros and Friendly Ghosts, American Journal of Mathematics, Vol. 124, No. 1 (Feb., 2002), pp. 1-48 ...