"Birch and Swinnerton Dyer conjecture" usually refers to an amazing formula that predicts exactly the leading term of the L-function at $s=1$ (a real number $c$ and an integer $k$ such that the leading term is $c(s-1)^k$).
My question is
Why we consider with $s$＝1? Is there any necessity to think with $s$ = 1? Or is it just $s$ = 1 due to the accidental factor that surprising results are achieved when $s$ = 1?
If $n$ is any integer except 1, then the order of vanishing of $L(E, s)$ at $s = n$ is given by a simple formula (it's 0 if $n > 1$, and 1 if $n < 1$). This follows easily from the functional equation satisfied by $L(E, s)$. So the point $s = 1$, the center of symmetry of the functional equation, is the only integer where the order of vanishing isn't obvious, and thus has the chance to be related to some other interesting and nonobvious things, like the rank.
Incidentally, there's a more general conjecture, the Beilinson--Bloch--Kato conjecture, of which the BSD conjecture is a special case; and this conjecture also has something to say about the $L$-function at $s = n$ for any integer $n$. For $n \ne 1$ the order of vanishing is known but there's still lots of really interesting information encoded in the leading term of the $L$-function at $n$ (involving algebraic $K$-theory and Galois cohomology of $E$).