We have investigated analytically and numerically the liquid-glass transition of hard spheres for dimensions d-->infinity in the framework of mode-coupling theory. The numerical results for the critical collective and self-nonergodicity parameters fc(k;d) and fc(s)(k;d) exhibit non-Gaussian k dependence even up to d=800.fc(s)(k;d) and fc(k;d) differ for k approximately d1/2, but become identical on a scale k approximately d, which is proven analytically. The critical packing fraction phic(d) approximately d(2)2(-d) is above the corresponding Kauzmann packing fraction phiK(d) derived by a small cage expansion. Its quadratic pre-exponential factor is different from the linear one found earlier. The numerical values for the exponent parameter and therefore the critical exponents a and b depend on d, even for the largest values of d.