Today I was reading LMFDB (the L-functions and Modular Forms DataBase), and I came across something that confused me. When discussing degree 3 L functions on this page, they assert that all the ones found so far have Euler products of the form

$$L(s)=prod_{p|N}left(1-a_np^{-s}+left(a_p^2-a_{p^2}right)p^{-2s}right)^{-1}prod_{pnmid N}left(1-a_pp^{-s}+chi(p)overline{a_p}p^{-2s}-chi(p)p^{-3s}right)^{-1}$$

What I do not understand is why they chose to write $a_p^2-a_{p^2}$ instead of simply $a_{p^2}$. There is not other reference to $a_{n}$ anywhere else on the page and no information is given about $a_n$, and so I assume they are meant as arbitrary complex numbers, and so writing $a_{p^2}$ instead of $a_p^2-a_{p^2}$ would be just as complete and lose no generality. This hints to me that perhaps there are some restrictions (say, $Re(a_n)>0$) that are not being stated which calls for such a statement. Perhaps there is also “moral” reason to write it this way.

I note also that this is not an isolated phenominon on the LMFDB website. On the page for degree 4 L functions here they assert that all known L functions of degree four have Euler products of the form

$$L(s)=

prod_{p|N} left(1-a_p, p^{-s} + (a_p^2 – a_{p^2}), p^{-2s} – (a_p^3 – 2 , a_{p^2} , a_p + a_{p^3} ) , p^{-3s}right)^{-1}cdotprod_{pnmid N} left(1-a_p, p^{-s} + (a_p^2 – a_{p^2}), p^{-2s} – chi(p) , overline{a_p} , p^{-3s} +chi(p) , p^{-4s}right)^{-1}

$$

which one again uses the notation $a_p^2-a_{p^2}$, but now also uses the expression $a_p^3-2a_{p^2}a_p+a_{p^3}$ instead of $a_{p^3}$ which would lose no generality. Any insights are appreciated.