Central $L$ values of congruent number elliptic curves

Let $E_n$ be the congruent number elliptic curve $y^2=x^3-n^{2}x$, where $n$ is square-free and not divisible by primes $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. In this paper, we prove that $L(E_n,1)$ can be expressed as the square of CM values of some simple theta functions, generalizing two classical formulas of Gauss. Our result is meaningful in both theory and practical computation.