Let me summarise the comments above that give a full answer (correct me if I am wrong).
The analytic continuation of $L(E,\bar{s})=\overline{L(E,s)}$ shows that $c\in\mathbb{R}$.
That $c>0$ is proven in On the positivity of the central value of automorphic L-functions for GL(2) Duke Math 83 1996, 1-18. It would follow from the generalised Riemann hypothesis, too.
That $L(E,1)/\Omega_E$ is a rational number is a consequence of the theorem of Manin-Drinfeld on modular symbols.
Is just 2+3. Note that $\Omega_E$ is defined to be positive, as it is the least positive real period of a Néron differential on $E$ (or twice that depending on your normalisation).