See here for the answer (in particular, it's in section 1 and 2). Basically, the answer isn't pretty - it depends on the local components of the automorphic representation $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ associated to the Maass form $f$.

So if $p \nmid N$, then the local component $\pi_p$ of $\pi$ is a principal series representation $\omega_{p,1} \boxplus \omega_{p,2}$, where $\omega_{p,1}, \omega_{p,2}$ are unramified characters of $\mathbb{Q}_p^{\times}$. We have that
\[L_p(s,\pi_p) = \frac{1}{(1 - \omega_{p,1}(p) p^{-s})(1 - \omega_{p,2}(p) p^{-s})}\]
where $\omega_{p,1}(p) + \omega_{p,2}(p) = \lambda_f(p)$, the Hecke eigenvalue of $f$ at $p$, and $\omega_{p,1}(p) \omega_{p,2}(p) = \chi(p)$. Then the local Rankin-Selberg $L$-function is
\[L_p(s,\pi_p \times \widetilde{\pi_p}) = \frac{1}{(1 - |\omega_{p,1}(p)|^2 p^{-s})(1 - \omega_{p,1}(p) \overline{\omega_{p,2}(p)} p^{-s})(1 - \overline{\omega_{p,1}(p)} \omega_{p,2}(p) p^{-s})(1 - |\omega_{p,2}(p)|^2 p^{-s})}.\]

If $p \mid N$ and $\pi_p$ is a principal series representation $\omega_{p,1} \boxplus \omega_{p,2}$, where at least one of $\omega_{p,1}, \omega_{p,2}$ is ramified, then the same result holds as for the case $p \nmid N$ with the minor change that if $|\omega_{p,1}|^2$ is a ramified character of $\mathbb{Q}_p^{\times}$, we replace $|\omega_{p,1}(p)|^2$ with $0$, and similarly if $\omega_{p,1} \overline{\omega_{p,2}}$, $\overline{\omega_{p,1}} \omega_{p,2}$, or $|\omega_{p,2}|^2$ is ramified (see Jeremy's comment below). Note that if both $\omega_{p,1}$ and $\omega_{p,2}$ are ramified then $p^2 \mid N$. This is part of Proposition 1.4 of the linked paper.

If $p \mid N$ and the local component $\pi_p$ of the automorphic representation is supercuspidal (which can only happen if $p^2 \mid N$), then $\lambda_f(p) = 0$ and
\[L_p(s,\pi_p) = 1,\]
and either
\[L_p(s,\pi_p \times \widetilde{\pi_p}) = \frac{1}{1 - p^{-s}}\]
or
\[L_p(s,\pi_p \times \widetilde{\pi_p}) = \frac{1}{1 - p^{-2s}}.\]
The former occurs if $\pi_p \ncong \pi_p \otimes \eta_p$, where $\eta_p$ is the unramified quadratic character of $\mathbb{Q}_p^{\times}$, while the latter occurs if $\pi_p \cong \pi_p \otimes \eta_p$. This is Corollary 1.3 of the linked paper.

If $p \mid N$ and the local component $\pi_p$ is a Steinberg representation $\omega_p \mathrm{St}_p$, where $\omega_p$ is a unitary character of $\mathbb{Q}_p^{\times}$, then $\widetilde{\pi_p} = \omega_p^{-1} \mathrm{St}_p$. We have that
\[L_p(s,\pi_p) = L_p\left(s + \frac{1}{2}, \omega_p\right)\]
and
\[L_p(s,\pi_p \times \widetilde{\pi_p}) = \zeta_p(s + 1) \zeta_p(s) = \frac{1}{(1 - p^{-s-1})(1 - p^{-s})}.\]
This is part of Proposition 1.4. Note that if $\omega_p$ is unramified, we must have that $p \parallel N$, $\lambda_f(p) = \omega_p(p) p^{-1/2}$, and so
\[L_p(s,\pi_p) = \frac{1}{1 - \omega_p(p) p^{-s - 1/2}},\]
If $\omega_p$ is ramified, then $p^2 \mid N$, $\lambda_f(p) = 0$, and
\[L_p(s,\pi_p) = 1.\]
In particular, if $N$ is squarefree and $\chi$ is the principal character, then for $p \mid N$, $\pi_p$ must be $\omega_p \mathrm{St}_p$ with $\omega_p$ the trivial character, so $\lambda_f(p) = p^{-1/2}$.

Finally, the archimedean $L$-function is
\[L_{\infty}(s,\pi_{\infty}) = \pi^{-s - \kappa} \Gamma\left(\frac{s + i\mu + \kappa}{2}\right) \Gamma\left(\frac{s - i\mu + \kappa}{2}\right),\]
where $\kappa = 0$ if $f$ is even and $1$ if $f$ is odd, and
\[L_{\infty}(s,\pi_{\infty} \times \widetilde{\pi_{\infty}}) = \pi^{-2s} \Gamma\left(\frac{s + 2i\mu}{2}\right) \Gamma\left(\frac{s}{2}\right)^2 \Gamma\left(\frac{s - 2i\mu}{2}\right).\]

The completed $L$-function $\Lambda(s,f \times \widetilde{f})$ is then defined to be
\[\Lambda(s,f \times \widetilde{f}) = L_{\infty}(s,\pi_{\infty} \times \widetilde{\pi_{\infty}}) \prod_p L_p(s,\pi_p \times \widetilde{\pi_p}).\]
Theorem 2.2 of the linked paper states that this extends meromorphically to the entire complex plane with only simple poles at $s = 0$ and $s = 1$.