Eisenstein Series

$$E_r(t)=\setbox0=\hbox{$\scriptstyle{m, n}$}\setbox2=\hbox{$\... ...fi\fi \mathop{{\sum}'}_{\kern-\wd4 m, n} {1\over (mt+n)^{2r}},$$

where the sum $\Sigma'$ excludes $m=n=0$, $\Im[t]>0$, and $r$ is an Integer $>2$. The Eisenstein series satisfies the remarkable property

$$E_r\left({at+b\over ct+d}\right)= (ct+d)^{2r}E_r(t).$$

See also Ramanujan-Eisenstein Series