It is shown that the redshifts z(o) of gamma-ray burst (GRB) sources, where they have been measured, together with the redshifts for seven quasars (QSOs) that lie very close to the positions of the unidentified sources GRB 990625, 000210, 001105 (two QSOs), 940720, 991217, and 990506, show a remarkable tendency to cluster about several of the periodic redshift peaks previously established for QSOs at z=0.061, 0.30, 0.60, 0.96, 1.41, 1.96, 2.63, 3.44, and 4.45. In 1971, Karlsson showed that these peaks lie in a series with Deltalog (1+z)=0.089. Out of a total of 32 currently known redshifts of GRBs, afterglows, or QSOs very close to burst positions, two are very close to 0.30, three are close to 0.60, nine are equal to or very close to 0.96, three are very close to 1.41, six are close to 1.96, two are close to 3.44, and one is very close to 4.45. Statistical tests by W. Napier show that the observed redshifts z(o) showed periodicity at the 98% confidence level. In addition, very close to the positions of two bursts GRB 990625 and GRB 001105, many QSOs with redshifts close to the peak values have been found. Since z(o)=[(1+z(c))(1+z(D))(1+z(i))-1], where z(c), z(D), and z(i) are the cosmological, Doppler, and intrinsic components of the observed redshift zo, the existence of these peaks suggests that z(o)similar or equal toz(i), so that both z(c) and z(D) are very much less than z(o). However, while the observed values of z(o) are very close to the corresponding values of z(i), in most cases z(o)>z(i), suggesting that in most cases z(c) is greater than it was found to be in earlier samples of X-ray QSOs that appear to be ejected from bright galaxies. It appears likely, therefore, that the GRB sources, like the QSOs, are ejected from active galaxies, most of which have comparatively small cosmological redshifts 0.02less than or equal toz(c)less than or equal to0.1, thus suggesting that the distances of most of the GRB sources are less than or equal to500 Mpc. A possible example of an active galaxy that has given rise to such phenomena is UGC 12348 (z(i)=0.03). This galaxy has two GRB sources and three QSOs with measured redshifts z(o) that lie very close to intrinsic redshift peaks all lying within 1degrees of it. Among these five objects, the QSO at z=3.7 lies inside the error box for the unidentified burst GRB991217.