Let's the zeta function is defined as $$ \zeta(s)= \prod_p (1-p^{-s})^{-1}=\prod_p \frac{1}{(1-p^{-s})} $$ so for a negative therm should be $$ \zeta(-s)= \prod_p (1-p^{s})^{-1}=\prod_p \frac{1}{(1-p^{s})} $$ for the Euler product where $p$ is prime and the product is over all the primes. I can say that the productorial therm could be rewritten as: $$ (1-p^{-s})=(1-\frac{1}{p^s})=(\frac{p^s-1}{p^s})=(\frac{1-p^s}{-p^s}) $$ So computing the ration between $\zeta(s)$ and $\zeta(-s)$ will result in
$$ \frac{\zeta(s)}{\zeta(-s)}=\frac{\prod_p (1-p^{s})}{\prod_p (1-p^{-s})}=\prod_p\frac{ (1-p^{s})}{ (1-p^{-s})} =\prod_p\frac{ (1-p^{s}) (-p^s)}{ (1-p^{s})}= \prod_p -p^s $$
That is, the ratio of the zeta function with positive and negative argument $s$ should be equal to the negated complex power $s$ of all the primes.
This is a weird result, but I cannot understand what's wrong in here and especially the meaning of this result.
Sorry for the newbie (mathematical) question.
The analytic continuation of $\zeta(s)$ Dirichlet series and Euler product is exactly the same idea as for the geometric series $$\frac1{1-z}=\sum_{n\ge 0} z^n$$ which is valid only for $|z|<1$.
Your same reasonning would lead to
$$0=\frac1{1-5}+\frac18\frac{1}{1-1/2}= \sum_{n\ge 0} (5^n+\frac18 (1/2)^n)$$ which is obviously wrong.