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 $\alpha_f(p),\beta_f(p)$ (resp. $\alpha_g(p),\beta_g(p)$) be 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.