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


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



Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?
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. 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)](http://projecteuclid.org/euclid.dmj/1077244251) 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 $r>1$.
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).


