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.

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 $$\sum_{\rho}\dfrac{1}{|\rho|^2}=\dfrac{2}{k}\dfrac{\Lambda'(k)}{\Lambda(k)}$$ (add $2/k^2$ and remove polar part if noncuspidal). Thus, in the case of $\Delta$ the sum is $$0.101406744829750588788908230295355490988...$$ Many similar sums can be computed in the same way.