The document discusses the Legendre polynomial and the Birch and Swinnerton-Dyer conjecture. It provides equations for the Riemann zeta function L(C,s) and a second order linear differential equation related to Legendre polynomials. It also gives expressions for the eigenvalues λ and coefficients cn of the Legendre polynomial series expansion.

1. The Legendre Polynomial Of The Birch and Swinnerton-
Dyer Conjecture
Gregory M. Prew
8/20/2015

2. L(C,s) = c(s-1)r
s=1, r=2, & c≠0
= c(s-1)(s-1) = c(s2
-2s+1)
(1-s2
)y´´(s) - 2sy´(s) + λ y(s) = 0
∞
y1 = c0 + ∑ c2ks2k
k=1
λ = n(n+1) = 2(2+1) = 6
cn = (2n)!
2n
(n!)2
y1 = 1+ 3/2 (1)2
+ 4.375 (1)4
+…+cnsn