Why are negative even integers zeros for the Riemann Zeta Function?

Question

First I would just like to clarify I understand that the Riemann Zeta Function is only defined for $s>1$ and that we have to use other formulas that use analytic continuation to "define" the other values of $s$. What I don't understand however is why does the analytically continued function give us zero when $s$ is a negative even integer. I couldn't seem to find any proofs online and I would love to know the answer.