Let
E
E
be an elliptic curve over
Q
\mathbb {Q}
with discriminant
Δ
E
\Delta _E
. For primes
p
p
of good reduction, let
N
p
N_p
be the number of points modulo
p
p
and write
N
p
=
p
+
1
−
a
p
N_p=p+1-a_p
. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies
lim
x
→
∞
1
log
x
∑
p
≤
x
p
∤
Δ
E
a
p
log
p
p
=
−
r
+
1
2
,
\begin{equation*} \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}, \end{equation*}
where
r
r
is the order of the zero of the
L
L
-function
L
E
(
s
)
L_{E}(s)
of
E
E
at
s
=
1
s=1
, which is predicted to be the Mordell-Weil rank of
E
(
Q
)
E(\mathbb {Q})
. We show that if the above limit exits, then the limit equals
−
r
+
1
/
2
-r+1/2
. We also relate this to Nagao’s conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit.