An identity involving Weierstrass $\zeta$ function.

Question

The following identity is verified by Mathematica and PARI/GP: $$ 2\zeta\big(\frac{1+a}{2}\big)\big[=2\zeta\big(\frac{1}{2}\big) + 2\zeta\big(\frac{a}{2}\big)\big]= 3\zeta\big(\frac{1+a}{3}\big), $$ where $a=e^{i\pi/3}$ and $$ \zeta(z)=\frac{1}{z}+\sum_{0\ne w \in \left< 1,a\right>}\Big(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\Big). $$ is the Weierstrass $\zeta$ function. Is there a way to see this identity from the properties of $\zeta$ without any computation?

Answer 1

OK, I solved this by myself.

Consider the integrals of $\zeta'(z)=-\wp(z)$ along the segments $[1/2,(1+a)/3]$, $[a/2, (1+a)/3]$ and $[(1+a)/2, (1+a)/3]$. They sum up to $0$ because of the symmetry of the lattice.