Relationship between Eisenstein series $E_2$ and Eisenstein series $G_2$

Question

One can find two series called Eisenstein series of weight 2. The first one is $$ E_2(\tau) = 1 - 24 \sum_{n\geqslant 1}\frac{n q^{2n}}{1-q^{2n}} \quad \textrm{with } \; q=\exp(i\pi\tau) $$ and the second one is $$ G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}}\frac{1_{(c,d)\ne (0,0)}}{(c\tau+d)^2}. $$ What is the link between these two series? I thought this is $$ G_2 = \frac{\pi^2}{3} E_2, $$ but that does not fit to my numerical checks.