Concrete Example of the Birch and Swinnerton-Dyer Conjecture The Setup
Consider an elliptic curve $E$ in Weierstrass form
$y^2=x^3+ax+b$
with $a,b \in \mathbb{Z}$. As usual, we let $\Delta_E$ be the discriminant of the polynomial, and we set
$N_p := $ #{solutions of $y^2 \equiv x^3+ax+b$ mod $p$} 
$a_p := p-N_p. $
Now we can define the incomplete $L$-series of $E$ (incomplete because we onit the Euler factors for primes $p|2\Delta$) by
$L(C,s) := \prod _{p \nmid \Delta_E} (1-a_p ^{-s} + p^{1-2s})^{-1}.$
The Birch and Swinnerton-Dyer conjecture states that $L(C,s)=c(s-1)^r +$ higher order terms.
Where $r$ is the rank of $E$
The Question
My question is can someone show me a concrete example of the BSD conjecture in action? Specifically, could someone give me a specific elliptic curve, show me why the curve algebraically has the rank it has, and what the L-function of the elliptic curve is and how the analytic rank equals the algebraic rank?
Thanks!

You should read
On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
J.P. Buhler,  B.H. Gross and D.B. Zagier
Mathematics of Computation vol 44 1985 pp 473-481
You should be able to download it free from the AMS web-site
