# Are the d quantities log Gamma(\lambda_j.s+\mu_j) linearly independent over Q for all s>1?

This question deals with the gamma factor of a primitive function of the Selberg class. Writing the functional equation of such a function $F$ as $\Phi(s)=\overline{\Phi(\overline{1-s})}$ with $\Phi(s):=\gamma(s)F(S)$, $\gamma(s):=e^{i\phi}Q^{s}\prod_{j=1}^{d}\Gamma(\lambda_{j}s+\mu_{j})$ with $\phi$ real, $Q$ real and positive, the $\lambda_{j}$ real and positive and $\mu_{j}$ complex with non-negative real part, is it true that the quantities $\log\Gamma(\lambda_j.s+\mu_j)$ are linearly independent over $\mathbb{Q}$ for all $s>1$?