Theorem 1   Let $k$ be an even integer and let $B_k$ be the $k$ th Bernoulli number. Let $\zeta(s)$ be the Riemann zeta function. Then: $$\zeta(k)=\frac{2^{k-1}|B_k|\pi^k}{k!}$$ Moreover, by using the functional equation , one calculates for all $n\geq 1$ : $$\zeta(1-n)=\frac{(-1)^{n+1}B_n}{n}$$ which shows that $\zeta(1-n)=0$ for $n\geq 3$ odd. For $k\geq 2$ even, one has: $$\zeta(1-k)=-\frac{B_k}{k}.$$

Remark 1   The zeroes of the zeta function shown above, $\zeta(1-n)=0$ for $n\geq 3$ odd, are usually called the trivial zeroes of the Riemann zeta function, while the non-trivial zeroes are those in the critical strip.