Let me summarise the comments above that give a full answer (correct me if I am wrong).

1. The analytic continuation of $L(E,\bar{s})=\overline{L(E,s)}$ shows that $c\in\mathbb{R}$.

2. That $c>0$ is proven in <a href="http://projecteuclid.org/euclid.dmj/1077244251">On the positivity of the central value of automorphic L-functions for GL(2)</a> Duke Math 83 1996, 1-18. It would follow from the generalised Riemann hypothesis, too.

3. That $L(E,1)/\Omega_E$ is a rational number is a consequence of the theorem of Manin-Drinfeld on modular symbols. 

4. 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).