Infinity factor of an L-function (awaiting review)

The infinity factor $L_{\infty}(s)$ of an L-function $L(s)$ is the product of its Gamma factors: \[ L_{\infty}(s) := \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k). \] Here the $\mu_j$ and $\nu_k$ are the spectral parameters of the L-function.