It is well-known that Euler calculated the values of $\zeta(2n)$ for $n>0$. However, it is less known that he essentially discovered the functional equation of the Riemann zeta function. Euler was interested in the divergent sums
$$1+2^n+3^n + \dots$$
when $n>0$. He noticed that
$$\frac{t}{1+t} = t-t^2+t^3 - \dots$$
so, with $t=1$ we find $$1-1+1-1+\dots = \frac{1}{2},$$
an answer which is of course very silly if we take the left hand side literally. But $1/2$ is indeed the "average" of the partial sums on the left hand side, so maybe it is not completely silly. If we continue with this insanity we find:
$$t\frac{d}{dt}\frac{t}{1+t} = \frac{t}{(1+t)^2} = t - 2t^2 + 3t^3 - \dots$$
and therefore $$1-2+3-\dots = \frac{1}{4}.$$ In general, we have
$$\left(t\frac{d}{dt}\right)^n\frac{t}{1+t}\biggr|_{t=1} = 1-2^n+3^n-4^n+\dots.$$
On the other hand, if we put $t=e^x$, then $t \frac{d}{dt} = \frac{d}{dx}$, and $t=1$ becomes $x=0$:
$$\frac{d^n}{dx^n}\frac{e^x}{1+e^x}\biggr|_{x=0} = 1-2^n+3^n-4^n+\dots.$$
Now for $s>1$ a simple calculation shows that
$$1-2^{-s} + 3^{-s} - \dots = (1-2\times 2^{-s})\zeta(s),$$ hence in analogy, he defined $\zeta(-n)$ as
$$\zeta(-n):= (1-2^{n+1})^{-1} \frac{d^n}{dx^n}\frac{e^x}{1+e^x}\biggr|_{x=0}$$
For instance, for $n=1$ we find
$$\zeta(-1) = (1-4)^{-1} \frac{1}{4} = -1/12.$$
Anyways, it turns out that $e^x/(1+e^x)$ is essentially the generating function of the Bernoulli numbers, and after some simple manipulations, Euler obtained that for $n>1$,
$$\zeta(1-n) = -\frac{B_n}{n}.$$
(Notice in passing that this reveals the "trivial zeroes" of the Riemann zeta function (since the odd Bernoulli numbers vanish).) Hence, putting this together with his work on the values $\zeta(2n)$, Euler found the following explicit formulas:
$$\zeta(1-2n) = -\frac{B_{2n}}{2n}$$
$$\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}.$$
This is already enough to conjecture the form the functional equation should take, and Euler had all of the ingredients for it. However, it seems that Riemann was the first to explicitly write it down (and to prove it).
(I picked this up from Hida's book Elementary theory of $L$-functions and Eisenstein series (which is, for the most part, far from being elementary...).)
