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I am wondering how the relation is between of the automorphic L-function and its lift's.

More precisely,

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over E of dimension $n$, and let $V$ be a skew-hermitian space of dimension $m$ over $E$. Consider the associated unitary groups $H:=U(W)$ and $G:=U(V)$. Let $σ$ be an irreducible, cuspidal, automorphic representation of $H(A_F)$. Let $π=Θ(σ,ψ,γ)$ be a theta lift of $σ$ to $G(A_F)$. ($ψ:A_{F}/F→C^×$ and $γ:A_E^\times/E^{\times}→C^{\times}$ are the splitting data necessary to define the theta-lift for unitary groups.)

When $|m-n|\le1$ , it is conjectured and for small dimension case was discussed in chap 7 of

In the paper, it conjectures that the L-parameter of $\theta(\sigma)$, denoted by M, is tied to N, the L-parameter of $\sigma$ by $M=N$ for $m=n$ and $M=\mu^{-1} N \oplus \mu^{m}$ for $m=n-1$

But my primary concern lies in the case $m=n+2$, especially $n=1$ and $m=3$ or $n=2$ and $m=4$

Do you know the general rule for this case $m=n+2$ or even in the above special cases? If you know the answer or some reference regarding this and would you let me know? I will be very happy and grateful for your warm heart.

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