Artin's constant (used in his conjecture on primitive roots), is defined as
$$\mathrm{C_{Artin}} = \prod_{p \ \mathrm{primes}}{\left(1-\frac{1}{p(p-1)}\right)} = 0.3739558136\dots$$
(A005596 in the OEIS)
Are there any methods that converge faster than this?