Multi-antenna spectrum sensing algorithms for cognitive radio are receiving a lot of attention recently. In this paper, we consider multi-antenna detection when the noise covariance matrix is assumed to be arbitrary and unknown. The studies leading to this paper have been motivated by the existence but typically unknown noise correlation in practice. A multiple primary user detector, derived from the generalized likelihood ratio criterion, is analyzed in such a scenario. We calculate the exact moments of the test statistics involved, which lead to a simple and accurate analytical formula for the false alarm probability. The result is obtained by utilizing tools from multivariate analysis as well as moment based approximations. Simulations are conducted to examine accuracy of the derived result, with the achieved accuracy being reasonably good. From the considered simulation settings, performance gain over existing detection algorithms is observed in scenarios with arbitrary but unknown noise correlation and multiple primary users.