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 power, 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 $4n$ 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 $c$ are number of solutions of some cubic equations over $\mathbb{F}_n$, i.e., numbers of points of some curves over $\mathbb{F}_n$. For example, $a$ is the number of solutions of the equation 
\begin{equation}\label{final}
x^3-y^3=1.
\end{equation}
Suppose now that $p$ is an odd prime such that $p\equiv 1\pmod{3}$. Then $p^r\equiv 1\pmod{3}$ for all integers $r>0$. Then the above argument, in principle, determines all the coefficients $N_{\nu}$ 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][1], 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 $u$?


  [1]: https://en.wikipedia.org/wiki/Local_zeta-function#Examples