Mikael Passare showed how to compute $\zeta(2)$ (_[How to compute $\sum 1/n^2$ by solving triangles](https://arxiv.org/abs/math/0701039)_) using the [amoeba](http://www.ams.org/notices/200208/what-is.pdf) of $1 + z + w = 0$.  Has this ever been generalized to higher zeta-values?  How might one compute $\zeta(2n)$ this way?

$$ \zeta(2) = \int_0^\infty \log(1 - e^{-x}) \, dx = \frac{1}{3}\int_{s + t < \pi} 1 \; ds \, dt $$

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/0hecL.png