I'm reading Gelbart's classic (Automorphic forms on adeles groups) in a seminar. And now we're dealing with section 6.C, Jacquet-Langlands's work using local-global zeta integral as follows: $$ Z(W, s, \chi, g)=\int_{F^{\times}} W\left(\left(\begin{array}{cc} y & 0 \\ 0 & 1 \end{array}\right) g\right) \chi(y)|y|^{s-1 / 2} d^{\times} y $$ So I understand what is the $L$-factor and $\epsilon$-factor now, they're the g.c.d of common denominators and the factor in functional equation. Now my professor asked me to try to compute part of the 2 tables on Gelbart page 113-114, at least for unramified cases.
He suggested that a good reference for this is Cogdell's many notes on integral representations of $L$-functions, and I read some of his fields lecture notes (cf. his homepage, it can be downloaded there). So in lecture 7, he used Hecke algebra and some other tools to obtain Shintani's general computation for $GL(n)\times GL(m)$. And Jacquet-Langlands is of course the concrete case $n=2,m=1$.(I can't follow that currently) What I need is: how to compute $$ Z(W^0, s, \chi, 1)=\int_{F^{\times}} W^0\left(\left(\begin{array}{cc} y & 0 \\ 0 & 1 \end{array}\right) \right) \chi(y)|y|^{s-1 / 2} d^{\times} y $$ for unramified $\chi$ and normalized unramified $W^0$ (i.e. fixed by $GL(2,\mathfrak{o})$ by right regular representation).
Can anyone give me some clue of the main idea of this computation? (I don't think I can read Jacquet-Langlands' original book...) Or even how to compute this in principal series case? Thanks for any help!!