$L$-functions for $\Theta$-lifts

Question

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over $E$ of dimension $2,$ and let $V$ be a skew-hermitian space of dimension $3$ over $E.$ Consider the associated unitary groups $H:=U(W)$ and $G:=U(V).$ Let $\sigma$ be an irreducible, cuspidal, automorphic representation of $H(\mathbb{A}_F).$ Let $\pi=\Theta(\sigma,\psi,\gamma)$ be a theta lift of $\sigma$ to $G(\mathbb{A}_F)$. ($\psi:\mathbb{A}_F/F\to \mathbb{C}^\times$ and $\gamma:\mathbb{A}_E^\times/E^\times\to\mathbb{C}^\times$ are the splitting data necessary to define the theta-lift for unitary groups.)

My question is, how do automorphic $L$-functions (standard, adjoint, etc.) for $\pi$ relate to those for $\sigma$?

Answer 1

This question is answered in a paper of Gan, Gross, and D. Prasad. Here's a link:

http://www.math.ucsd.edu/~wgan/ggp-evidence4-1.pdf

The relation between L-parameters of representations and their theta-lifts (at least locally) is discussed in section 7 of the paper.