  Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant $\Delta_E$.
For primes $p$ of good reduction, let $N_p$ be the number of points modulo $p$
and write $N_p=p+1-a_p$. In 1965, Birch and Swinnerton-Dyer formulated a
conjecture which implies $$\lim_{x\to\infty}\frac{1}{\log
x}\sum_{\substack{p\leq x\\ p\nmid \Delta_{E}}}\frac{a_p\log
p}{p}=-r+\frac{1}{2},$$ where $r$ is the order of the zero of the $L$-function
$L_{E}(s)$ of $E$ at $s=1$, which is predicted to be the Mordell-Weil rank of
$E(\mathbb{Q})$. We show that if the above limit exits, then the limit equals
$-r+1/2$. We also relate this to Nagao's conjecture.
