The family of hyperKloosterman sums can be studied through the k-th symmetric power L-function. These L-functions are p-adically continuous in k, and so we may take limits, producing the infinite symmetric power L-function. This work develops a p-adic cohomology theory and Frobenius structure for these L-functions, similar in nature to classical L-functions, except in this case we do not expect rationality. Instead, we expect H^1 to be infinite dimensional, and prove H^0 is at most of finite dimension or 0. Consequently, these L-functions are p-adic entire.