sequences and series - How to compute the Riemann zeta function at negative integers?

There already questions such as $1 + 1 + 1 +\cdots = -\frac{1}{2}$ and Why does $1+2+3+\cdots = -\frac{1}{12}$? which show how $\zeta(0)$ and $\zeta(-1)$ can be calculated.

What are some ways to evaluate the Riemann zeta function at any negative integer that appears to have a direct correlation to $\sum_{n=1}^\infty\frac1{n^s}$? That is to say, results that can be attained by manipulating this series somewhat directly. So not things such as the reflection formula or the Bernoulli numbers which don't seem to relate to the above series.