In this paper, we develop an algorithm for computing the zeros of a generalized state-space model described by the matrix 5-tuple (E, A, B, C, D), where E may be a singular matrix but det(A-lambda E) not equal 0. The characterization of these zeros is based on the system matrix of the corresponding 5-tuple. Both the characterization and the computational algorithm are extensions of equivalent results for state-space models described by the 4-tuples (A,B,C,D). We also extend these results to the computation of infinite zeros, and left and right minimal indices of the system matrix. Several non-trivial numerical examples are included to illustrate the proposed results.