This question is motivated by the work presented in article 358 of Gauss' *Disquisitiones Arithmeticae*. For the sake of completeness, let me say something about the background and present the question at the end. Any comment or correction is appreciated.

Let $n$ be an odd prime, and suppose further that $n=3m+1$ for some integer $m$. Let $g$ be a generator of the cyclic group $(\mathbb{F}_n)^{\times}$, and let $H$ be its unique subgroup of index $3$. In the above mentioned work, Gauss denotes the cosets $H$, $gH$ and $g^2H$ respectively by $R$, $R'$ and $R''$. He further denotes by $p$ (respectively, $p'$ and $p''$) the Gauss sums $$\sum_{w}\exp{2\pi iw},\textrm{ for $w$ running over $R$ (respectively, $R'$ and $R''$).}$$ He then studies the polynomial equation \begin{equation}\label{question} x^3-Ax^2+Bx-C=0 \end{equation} of which the roots are $p$, $p'$ and $p''$.

One easily find $A=-1$ since $A=p+p'+p''$ is the sum of all $n$-th roots of unity except $1$.

To determine $B$ and $C$, Gauss considers the set
$$RR:=\{w\in (\mathbb{F}_n)^{\times}|w\in R, w+1\in R'\}$$

and similarly defined sets $RR'$, $RR''$, $R'R''$, etc., and denotes by $(RR)$, $(RR')$, etc. their cardinalities. He further determines
\begin{equation}\label{abc}
\begin{cases}
(R'R'')=(R''R')=:a,\\
(R''R'')=(R'R)=(RR')=:b,\\
(R'R')=(R''R)=(RR'')=:c.
\end{cases}
\end{equation}
and
\begin{equation}\label{a+b+c}
a+b+c=m.
\end{equation}

To determine $B$, Gauss uses elementary yet sophisticated tricks to express the products $pp'$, $pp'$ and $p'p''$ in terms of linear combinations of $p$, $p'$ and $p''$ with coefficients $a$, $b$ and $c$. Then using a symmetry argument, Gauss proves $$B=m(p+p'+p'')=-m.$$

By various computational exertions, Gauss proves $$C=a^2-bc,$$ and \begin{equation}\label{long} a^2+b^2+c^2-a=ab+bc+ac, \end{equation} which yields \begin{equation}\label{4n} 4n=12a+12b+12c+4=(6a-3b-3c-2)^2+27(b-c)^2. \end{equation} Gauss then shows that the number $4n$ (for $n$ prime, or see @Will Sawin's comment) can be written in a unique way as $M^2+27N^2$ for integers $M$, $N$. Hence the values $a$, $b$, $c$, and therefore $C$, are completely determined.

Now notice that $a$, $b$ and $c$ give rise to numbers of solutions of some cubic equations over $\mathbb{F}_n$, i.e., numbers of points of some curves over $\mathbb{F}_n$. For example, $a+3=(RR)+4$ is the number of solutions of the equation \begin{equation}\label{final} x^3-y^3=1. \end{equation}

The above argument, in principle, determines all the coefficients $N_1$ of the local zeta-function of (the associated projective variety of) the above equation $$Z(u)=\exp{\sum_{\nu=1}^{\infty}N_{\nu}\frac{u^{\nu}}{\nu}}.$$

As mentioned in this wikipeida page, this is the earliest known non-trivial cases of local zeta-functions.

My question is: How does one (if possible) start from the above work of Gauss and express $Z(u)$ as a rational function in $u$?