We know that the Fourier transform for Riemann $\Xi$
${\displaystyle \Xi (t)=\xi ({\frac 12}+it) }$
where:
$\xi (s)={\tfrac {1}{2}}s(s-1)\pi ^{{-s/2}}\Gamma \left({\tfrac {1}{2}}s\right)\zeta (s)$
The Fourier transform of $\Xi(t)$ is:
$\Xi (t) = \int_{-\infty}^\infty\Phi(u)e^{iut}\,du$
Where:
$\Phi(u) = \sum_{n=1}^\infty (4\pi^2n^4e^{9u/2} - 6n^2\pi e^{5u/2} ) exp(-n^2\pi e^{2u})$
But, what are the Fourier transform for Dirichlet L function ?
$ {\Lambda(s,\chi) = \left(\frac{\pi}{k}\right)^{-(s+a)/2}\Gamma\left(\frac{s+a}{2}\right) L(s,\chi) }$
where symbol $a$:
$ {a=\begin{cases}0; \mbox{if }\chi(-1)=1, \\ 1;\mbox{if }\chi(-1)=-1,\end{cases}}$
Can anyone help with this question ? or can you provide a reference ?
(For example, for Dirichlet $\eta$ function, what is its Fourier Transform after consider $\Gamma(s/2)$ and $\pi ^{{-s/2}}$ adjustment ? )
Thank you for your attention.