We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values J randomly. Series for the Edwards-Anderson spin-glass susceptibility (EA) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent for d=4 and 5, are given. We quote =1.730.03, 2.000.25, and 2.7-0.6+1.0 and =0.950.04, 0.90.1, and 0.70.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of EA to measurements of derivatives of the magnetization with respect to a uniform magnetic field.