Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or other any such examples) \begin{align} &\int_{Z N_0 \setminus GSp_6} \int_{V^0} \int_{S^0} |W_\varphi[j( u(x_0) m(1,v_0) g)] f(g,s)| \, dx_0\, dv_0\, dg\\ &= \int_{K} \int_{Z\setminus T'} \int_{V_0} \int_{S^0} |W_\varphi[j( u(x_0) m(1,v_0) t' k)]| |a^2 b^4 c^8|^s \delta_{B_0}(t') \, dx_0 \, dv_0\, dt' \, dk. \end{align} where $K$ is the maximal standard compact subgroup of $GSp_6$ and $t'$ is over the maximal torus of $GSp_6$ modulo $Z$, where $Z$ is the center of $GSp_6$ (and also $GL_6$) and $\delta_{B_0}$ is the modular function of the Borel subgroup of $GSp_6$, following from the Iwasawa decomposition of $GSp_6 = Z N_0 T' K$ where $ZT'$ is a maximal torus of $GSp_6$ and $N_0$ consist of all upper unipotent matrices in $GSp_6$.
I am aware of Traces of Hecke Operators by Andrew Knightly and Charles Li, pp.102 that address some cases of quotient measure such as $$\int_{\mathbb{Q}^\times \setminus \mathbb{A}^\times} f(x) \, d^\times x = \int_{\mathbb{R}_{>0}} \int_{\widehat{\mathbb{Z}}^\times} f(r \times u) \frac{dr}{r}\, d^\times u$$ However, this involves the use of fundamental domain which seems very unwieldy.
Thank you,
