A certain infinite determinant arises as the constant in the block case of the Szego-Widom Limit Theorem for Toeplitz matrices. These are matrices whose block entries are constant on the diagonals. While the constant has a very nice form, often a more explicit form, analogous to the one found in the scalar case, is needed in applications. This talk will survey some of the ways the constant can be more explicitly computed (often using functional analysis techniques) and also some applications.