Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a partial L-function $L(s, \pi, r)$. However this one is defined only outside ramified places and archimedean places.
I am wondering about the definition of the "completed" version, adding these bad places. In the case of $GL(n)$ this is motivated essentially by completing the L-function in order to get a quite simple functional equation. I haven't found any proper reference for such a definition, is there something or is there a reason for avoiding to do so?
Is there a standard way to define a notion of archimedean L-factor using the archimedean Langlands classification (which would give a fairly general recipe for (almost?) every reductive group)?
