I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As usual, we denote by $\alpha_f(p),\beta_f(p)$ (resp. $\alpha_g(p),\beta_g(p)$)
 the local parameters of the attached $L$-function $L(f, s)$ (resp. $L(g, s)$) at a prime $p$. It is well-known that $$L(f\times g, s)=\sum_{n=1}^\infty  \frac{\lambda_{f\times g}(n)}{n^s}=\zeta^{(N)}(2s)\sum_{n=1}^\infty  \frac{\lambda_{f}(n)\lambda_{ g}(n)}{n^s},$$where $\zeta^{(N)}(2s)=\prod_{p\nmid N}(1-p^{-2s})^{-1}$. In addition, as far as I know, the local factors for $L(f\times g, s)$ which admits a Euler product $L(f\times g, s)=\prod_{p\nmid N}L_p(f\times g, s)$$\prod_{p\mid N}L^\prime_p(f\times g, s)$ is given by
\begin{align}L_p(f\times g, s)=&
\left (1-\frac{\alpha_f(p)\alpha_g(p)}{p^s} \right)^{-1}     
 \left (1-\frac{\beta_f(p)\alpha_g(p)}{p^s} \right)^{-1}  \\    
&\left (1-\frac{\beta_f(p)\beta_g(p)}{p^s} \right)^{-1}         
 \left (1-\frac{\beta_f(p)\beta_g(p)}{p^s} \right)^{-1};\end{align}
while, for $p|N$, 
$$L^\prime_p(f\times g, s)=\left (1-\frac{\lambda_f(p)\alpha_g(p)}{p^s} \right)^{-1}
\left (1-\frac{\lambda_f(p)\beta_g(p)}{p^s} \right)^{-1}.
$$



My question is how about the exact local factors for the $L$-function $$L(\pi, s)=\sum_{n=1}^\infty  \frac{\lambda^2_{f\times g}(n)}{n^s}?$$If one writes $L(\pi, s)=\prod_{p\nmid N}L_p(\pi, s)\prod_{p\mid N}L^\prime_p(\pi, s)$. How to write the two local factors $L_p(\pi, s)$ and $L^\prime_p(\pi, s)$ by means of the local parameters?

If any expert here leans some knowledge about this question, please help to show some guides or certain relevant references. Many many thanks.

Thanks in advance.

**EDIT:** According to David Loeffler's comment, the notation of this $L$-function is a bit misleading. So, probably one considers $\pi^\prime=f\times g$ and its associated Rankin-Selberg $L$-function 
$$L(\pi^\prime\times \widetilde{\pi^\prime},s)=\sum_{n=1}^\infty \frac{\lambda_{\pi^\prime\times \widetilde{\pi^\prime}}(n)}{n^s}$$instead, where $\widetilde{\pi^\prime}$ corresponds 
to the dual of ${\pi^\prime}$. This is of the Riemann type (i.e., satisfies certain functional equation of the
Riemann-type, and has analytic continuation to the whole complex plane, where it is holomorphic except
possibly for a pole at $s=1$), and particularly admits a factorization $$L(\pi^\prime\times \widetilde{\pi^\prime},s)=\prod_{p\nmid N}L_p(\pi^\prime\times \widetilde{\pi^\prime},s) \prod_{p\mid N}L^\prime_p(\pi^\prime\times \widetilde{\pi^\prime},s).$$So, my question reduces to how to determine the local factors of $L_p(\pi^\prime\times \widetilde{\pi^\prime},s) $ and $L^\prime_p(\pi^\prime\times \widetilde{\pi^\prime},s) $ by means of the local parameters of $f,g$? 

Much much obliged for any comments/answers from the so many experts here in MO-website. Thanks in advance.