\[ \large \displaystyle \lim_{n \to \infty} \frac {\displaystyle \prod_{r=0}^n { \left [ (2(r+1))^2 \ ((2r+1)^2 - x^2) \right ]} }{[2(n+1)]^2 \ [(2n+1)!]^2} \]

Denote the limit above as \(L(x) \). If the coefficient of \(x^{48} \) of \(L(x) \) is \(C\), what is the value of \( \left\lfloor { 10 }^{ 4 }\frac { 48! }{ 12! } C \right\rfloor \)?

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