I found that in many literature, they only define the L-function of Hilbert modular forms of parallel weight. Can we define the following L-function: \begin{equation} L(f;s_1,\cdots,s_n)=\int_0^{i\infty}\cdots\int_0^{i\infty}f(\tau_1,\cdots,\tau_n)\tau_1^{s_1}\cdots\tau_n^{s_n}\frac{d\tau_1}{\tau_1}\cdots \frac{d\tau_n}{\tau_n}. \end{equation} I am wondering if there is any literature about it. If such definition is well-defined, then does it have the functional equation like \begin{equation} L(f;s_1,\cdots,s_n)=C\cdot L(f;k_1-s_1,\cdots,k_n-s_n) \end{equation} for some constant $C$ which doesn't depend on $f$?
Thank you for your insights!