On the Wolfram Mathworld page for the Riemann zeta function (here), just after equation $20$, we have the sentence:
While this formula defines $\zeta(s)$ for only the right half-plane $\Re[s]>0$, equation (◇) can be used to analytically continue it to the rest of the complex plane.
What is "equation (◇)"? Does it mean the last equation mentioned? It's used at several points in the article, and my mathematical intuition isn't strong enough to to figure out the meaning from the context.
In this case it isn't the last equation mentioned, because we got equation (20) from (19) by rearranging the terms. The equation (◇) is:
$$\Gamma\left(\frac{s}{2}\right)\pi^{-\frac{s}{2}}\zeta(s)=\Gamma\left(\frac{1-s}{2}\right)\pi^{-\frac{-1-s}{2}}\zeta(1-s)$$ Which is the reflection formula for $\zeta(s)$.
Why? well, we can define a function $f(s)$ as the left hand side, and then $f(s)=f(1-s)$. So, if we know enough about the Gamma function (which we do), using this functional equation we can analytically continue $\zeta(s)$ to the complex plane for $s\neq 1.$