The gN(s) are of interest as approximations to the Riemann zeta function. Let s = or + it. In [1], it was shown that for t sufficiently large, gi(s) and g2(s) have their zeros on the critical line ar = 2. After encountering analytical difficulties in attempting to extend this theorem to further N, the calculations described below were undertaken. The results strongly suggest that for N > 3 one can expect to find infinitely many zeros off ar = 2 so that the theorem proved in [1] appears at its natural limit. For each N there is a region where gq(s) behaves similarly to t(s), and also a region where it behaves similarly to 2t(s). This empirical information should prove very useful for work along the lines of Rouch6's theorem, giving a condition for the Riemann hypothesis to be true in terms of the location of the zeros of gv(s). In Section 2, we give the theory of calculating the number of zeros of an analytic function within a closed curve when the information comes from a finite number of points on the curve. The theorem requires, for application in our case, an estimate for I 9N'(S) I, and this estimate is obtained. In Section 3, the method used for calculating X(s) is described, a difficulty being the calculation of r(s) for low values of t. Section 4 contains a discussion of the real function ZN(t) analogous to the Z(t) of the t-function. In Section, 5 the general organization of the calculations is described and Section 6 contains a discussion of the results. There are tables and figures of the zeros at the end.