Landau-Ramanujan Constant

Let $N$ be a natural number. Consider the equation

$$x^2+y^2=N$$

Let $L(N)$ denote the number of naturals $z\le N$ such that the equation $x^2+y^2=z$ has at least one integer solution $(x,y)$.The Laudau-Ramanujan constant is defined as the limit

$$\underset{N\to \mathrm{\infty }}{lim}\frac{L(N)\sqrt{\log (N)}}{N}\approx 0.76422365358922066299069873125$$