Given a lattice $L$ and an integer $n\ge4$ we can calculate a number — the Eisenstein series of the lattice:
$$G_{n}(L) = \sum _{\omega \in L, \omega \neq 0} \frac{1}{\omega ^{n}}.$$
How do we calculate the Eisenstein series of the lattice below?
$X_2 = w_1^{\mathbb Z} \odot w_2^{\mathbb Z},$ where $w_1=(a,1)$ and $w_2=(1,b)$ for some $a,b>0; a,b\neq 1,$ and $(x_1,y_1)\odot(x_2,y_2)=(x_1 x_2, y_1 y_2)$ and $w^{\mathbb Z} = \{ w^n \mid n \in \mathbb{Z} \}.$
Can we use the $G_n(L)$ or do we need something different?
For example:
$$G_{4}(\mathbb{Z}) = \sum _{n \neq 0} \frac{1}{n^{4}} = 2 \zeta (4) = 2 \frac{\pi ^{4}}{90}$$
I'm having trouble convincing myself that
$$G_4(X_2) = \sum _{n \neq 0} \frac{1}{n^{4}} = 2 \zeta (4) = 2 \frac{\pi ^{4}}{90}$$
What seems to be apparent is that $X_2\simeq \Bbb Z.$ In other words $X_2$ is a transported lattice under the $\exp$ mapping. This transport of structure allows one to pass information from the lattice $\Bbb Z$ to the lattice $X_2$.
A natural question is how to pass the Eisenstein series from $\Bbb Z$ to $X_2,$ which is my question.