complex number $u,$ nonnegative integer $n,$ and Schwartz function $\varphi.$ Since $Jf_s$ is just the Jacquet integral of an embedded ${{\mathrm{SL}}}_2$ we have an asymptotic expansion as in [@Wallach-RRG2 15.2]. This can be formulated as follows. Let $(z_n)_{n=1}^\infty$ be the sequence of complex numbers obtained by numbering the elements of $\{ 3s_0+2k: k = 0,1,2, \dots\} \cup \{ 1-3s_0+2k: k = 0,1,2 \dots\}$ in increasing order of real part. Then there is a sequence $(a_k)_{k=0}^\infty$ of continuous linear functionals $I(s_0, \chi)\to {\mathbb{C}}$ and a sequence $(A_k)_{k=1}^\infty$ of continuous functions $I(s_0, \chi) \to (0, \infty)$ such that for any non-negative integer $N$, $$\left|
Jf_{s_0}(t) - \sum_{k=0}^N a_k(f_{s_0}) t^{z_k} 
\right|\le A_{N+1}(f_{s_0}) t^{z_{N+1}},~ \forall t < 1.$$ Let $Z(\varphi, u, n)$ be defined by $$Z(\varphi, u, n)=\int_0^\infty \varphi(t) t^u (\log t)^n \, dt.$$ for $u > 0$ and by meromorphic continuation elsewhere. (Notice that $Z(\varphi u, 0)$ is a standard Tate zeta factor, while $Z(\varphi, u, n)$ is its $n$th derivative.) Then $$\int_0^{\infty} Jf_s(t) \varphi(t) t^u (\log t)^n \, dt
- \sum_{k=0}^n a_k(f_{s_0})
Z( \varphi, u+z_k, n) 
+E(f_{s_0}),$$ where $$|E(f_{s_0})| \le A_{N+1}(f_{s_0})
\int_0^\infty |\varphi(t)| t^{{{\mathrm{Re}}}(u+z_{N+1})} ( \log t)^n \, dt,$$ provided we choose $N$ sufficiently large to ensure that this last integral is 