TITLE: Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?
QUESTION [9 upvotes]: Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}_{s = 1} L(E, s)$, a value conjecturally equal to $\mathrm{dim}_{\mathbb{Q}} (E(\mathbb{Q}) \otimes_{\mathbb{Z}} \mathbb{Q})$. There is a $c \in \mathbb{C}^\times$ such that $L(E, s) \sim c \cdot (s - 1)^r$ as $s \rightarrow 1$. Birch and Swinnerton-Dyer predict a great deal about $c$, in particular, that

$c \in \mathbb{R}$,
$c \in \mathbb{R}_{> 0}$,
$c/\Omega_E \in \mathbb{Q}$ if $r = 0$, where $\Omega_E$ is the volume of $E(\mathbb{R})$ computed with respect to "the" Neron differential of $E$, and
$c/\Omega_E \in \mathbb{Q}_{> 0}$ if $r = 0$.

Which of these predictions are known to hold?
Partial answers and answers in special cases (e.g., in the case when $E$ has potential complex multiplication) are very welcome!

REPLY [12 votes]: 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 $c\in\mathbb{R}$.
If $r=0$, the fact 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. For $r=1$, it follows from Gross-Zagier and the case $r=0$. I don't know about $r>1$.
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).