We conjecture the form of the one-loop determinants for localized gauge theories with eight supersymmetries on d-dimensional spheres. Combining this with results for the localized action, we investigate the strong coupling behavior in the large N limit for a continuous range of d. In particular, we find the N dependence of the free energy for supersymmetric Yang-Mills with only a vector multiplet in 3<d<4 and for maximally supersymmetric Yang-Mills in 3< d<6. We also argue that this gives an

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... ective way to regularize divergences after localization in d=4 for N=2 gauge theories and d=6 for the maximally supersymmetric case.