I'm probably too optimistic but I expect $\Lambda_4$ to have a closed-form expression.

Sorry for being lazy; I explicited the summation.

Nice expression. I have the feeling that a series enables more efficiency for numerical computation so I would express that as follows: \begin{align}g_1(\xi_\rho,\xi_3) - g_0(\xi) &= \frac{2\pi}{\xi_\rho} \Im\left(\int_0^1 e^{i\xi_3\sqrt{1-t^2}}\frac{t^2}{\sqrt{1-t^2}}J_1(\xi_\rho t)\mathrm{d}t\right) \\ &= i\frac{2\pi}{\xi_\rho} \sum_{n=0}^{+\infty} (-1)^n\frac{(\xi_3)^{2n+1}}{(2n+1)!}\int_0^1 t^2(1-t^2)^nJ_1(\xi_\rho t)\mathrm{d}t \\ & = i\frac{2\pi\xi_3}{(\xi_\rho)^2} \sum_{n=0}^{+\infty} \left(-\frac{2(\xi_3)^2}{\xi_\rho}\right)^n\frac{n!}{(2n+1)!‌​}J_{n+2}(\xi_\rho) \\ \end{align}