{{\mathrm{GL}}}_2$.

Let $\delta_P$ be the modulus character of $P$. One can check that $\delta_P(m)=|\det(m)|^3$ for $m\in M$. Let $\chi$ be a character of $F^\times$, we consider the normalized induced representation \[defn of I(s,chi)\] $$I(s,\chi)={{\mathrm{Ind}}}_{P}^{G_2}(\chi_{s-1/2}),$$ where $\chi_s$ is the character of $M$ defined by $\chi(\det(m))\delta_P^s(m)$. Note that $$\chi_s(h(a,b))=\chi(a^2b)|a^2b|^{3s}.$$

Let $\tilde w=w_\beta w_\alpha w_\beta w_\alpha w_\beta$. Then $\tilde w$ represents the unique Weyl element such that $\tilde w(\alpha)>0$ but $\tilde w(U^\alpha)$ is in the opposite of $U$. Consider the standard intertwining operator $$M_{\tilde w}: I(s,\chi){\rightarrow}I(1-s,\chi^{-1})$$ which is defined by $$M_{\tilde w}(f_s)(g)=\int_{U^\alpha}f_s( \tilde wug)du.$$ By the general theory of intertwining operators, $M_{\tilde w}$ is absolutely convergent for ${{\mathrm{Re}}}(s)\gg0$ and can be meromorphically continued to all $s\in {{\mathbb {C}}}$.

An exact sequence for induced representations of Lg
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For each $\rho \in F^\times,$ we have the double coset decomposition $G_2=Pw_\beta H_\rho\cup P H_\rho$, see [@H12 Lemma 3]. The subgroup $T_0:= {\left\{{h(a,1): a\in {{\mathrm{GL}}}_1}\right\}}$ of $T$ is contained in $H_\rho$ for all $\rho$ and is the maximal $F$-split subtorus of $T_\rho$ when $\rho$ is not a square. We also have $$H_\rho\cap w_\beta P w_\beta^{-1}=N_{2,\rho}\cdot T_0,$$ for all $\rho.$ See [@H12 pp. 198-199]. Moreover we have the relation $$\label{eq2.1}
w_\beta h(a,1)=h(1,a)w_\beta.$$ On the other hand, recall 