This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation.

So, I post it hoping some benevolent one shed me a light.

Let $\pi$ be a local component of some global irr.cusp.unitary automorphic representation of $U(2)$ at split finite place.(i.e. $\pi$ is $GL(2)$ reps)

Then, there are three possible candidates for $\pi$ except for supercuspidal;

1)$B(\chi_{1} , \chi_{2})$ for $\chi_{1} \cdot \chi_{2}^{-1}=1$

2)$B(\chi_{1} , \chi_{2})$ for $\chi_{1} \cdot \chi_{2}^{-1} \ne 1$

3)irreducible quotient of $B(\chi\left\vert \cdot \right\vert^{\frac{1}{2}},\chi\left\vert \cdot \right\vert^{-\frac{1}{2}})$ for unitary character $\chi$.

(here, all $\chi, \chi_{1} , \chi_{2}$ are character of $GL(1)$ )

Then, for any character $\gamma$ of $GL(1)$,

what is $L(s,\pi \otimes \gamma)$ for the above three each $\pi$?

(here, L-function is local L-function and we consider $\gamma$ as $GL(2)$ character through determinant map)

Since my main concern lies in computing the order of zero or pole of the above $L$-function at $s=0$, if it is hard to write explicitly in ramified case, would you just inform me the result for each cases? Then, I am very grateful for your kindness.

(For beginner in this area, getting used to L-function calculation is quite difficult.)