Let $\pi$ be a cuspidal automorphic representation of $GL_2(\mathbb{A}_\mathbb{Q})$ with trivial central character. Then we can attach an L-function to $\pi$ which can be written as a Dirichlet series

$ L(s,\pi) = \sum_{m\in\mathbb{N}} \frac{\lambda_\pi(m)}{m^s}. $

Using Hecke operators one can prove the following relation on the coefficients

$ \lambda_\pi(m)\lambda_\pi(n) = \sum_{a|(m,n)} \lambda_\pi(a^{-2}mn). $

From this one can, in particular, derive the relation $\lambda_\pi(p)^2-\lambda_\pi(p^2) = 1$, where $p$ is prime and $L$ has been properly normalized. I need a similar formula for $\pi$ a representation of $GL(4)$ instead of $GL(2)$. Has this been calculated, either in the adelic setting, or for classical modular forms on $GL_4(\mathbb{R})$? Thanks!