The Hardy-Littlewood Conjecture F [1] involves the infinite product $$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$ where $\varpi$ ranges over the odd primes and $\left(\frac D\varpi\right)$ is the Legendre symbol.

Is there a good way to calculate this? The product converges very slowly, and none of the standard methods (Cohen-Villegas-Zagier, Wynn, etc.) seem to work because of the unpredictable sign changes.

Given that *D* is fixed, it suffices to calculate the partial products in various congruence classes; I don't know if this is a viable approach.

Another possibility: I've seen almost magical series acceleration with the zeta function, it may work here.

[1] G. H. Hardy, J. E. Littlewood. "Some of the problems of partitio numerorum III: On the expression of a large number as a sum of primes". *Acta Mathematica* **44** (1923), pp. 1-70.