Determinants of Laplacians and multiple gamma functions

In this paper we generalize the classical formula $\Gamma (\frac{1}{2}) = \sqrt \pi $. We do this by recalling the Multiple Gamma Function first studied in the nineteenth century by Barnes and others. These functions at $\frac{1}{2}$ will be expressed in terms of the functional determinant of Laplacians of the n-sphere (thus these invariants of the n-sphere generalize $\pi $). Determinants of Laplacians have been a recent subject of research due to their relevance to Superstring Theory. While the determinant of the Laplacian has been computed for a flat torus using the Kronecker Limit Formula, our result gives the case of the n-sphere with the standard metric.