nt.number theory - $L$-functions for $Theta$-lifts

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 $G(mathbb{A}_F)$. ($psi:mathbb{A}_F/Fto mathbb{C}^times$ and $gamma:mathbb{A}_E^times/E^timestomathbb{C}^times$ 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 $sigma$?