Assuming the GRH for L-functions attached to modular forms, if $D$ is a cuspidal eigenform of weight $k$ with rational coefficients such as $\Delta$
and $\Lambda(s)=(2\pi)^{-s}\Gamma(s)L(s)$ the completed $L$-function
(so that $\Lambda(k-s)=(-1)^{k/2}\Lambda(s)$), then the Hadamard product
immediately implies that
$$\sum_{\rho}\dfrac{1}{|\rho|^2}=2\dfrac{\Lambda'(k)}{\Lambda(k)}$$
(add $2/k$ if noncuspidal). Thus, in the case of $\Delta$ the sum is
$$1.2168809379570070654668987635442658919...$$
Many similar sums can be computed in the same way.