Draft:Hardy-Littlewood constant

The Hardy-Littlewood constant of a given polynomial with integer coefficient is a constant related to how often this polynomial gives prime numbers, for example, the Hardy-Littlewood constant of the polynomial $$x^2+x+41$$ is 3.319773177471…, and the Hardy-Littlewood constant of the polynomial $$x^2+1$$ is 0.686406731409… , and the Hardy-Littlewood constant of the polynomial $$x^4+1$$ is 0.669740969937….

Records
The numbers k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant is

1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757, …

The numbers k > 0 such that the polynomial x^3 + x^2 + k produces a record of its Hardy-Littlewood Constant is

1, 17, 101, 1487, 13301, 19421, 91127, …

The numbers k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant is

1, 2, 3, 4, 12, 18, 28, 58, 190, 462, 708, 5460, 10602, 39292, 141100, 249582, 288502, …

Generalization
The Hardy-Littlewood constant for more than one polynomial (that give prime numbers simultaneously), for example, the Hardy-Littlewood constant for x^4+1 and (x+2)^4+1 both primes is 0.792208238167…