The value of Taniguchi's constant is 0.6782344. It can be derived as
CTaniguchi = pi[1-3/p3+2/p4+1/p5-1/p6]
To get the closed form of Taniguchi's constant, expand the sum till infinity by taking the logarithm.
CTaniguchi = exp[∞Σn=3 cn P (n) ].
Here P (n) is the prime zeta function and cn is the rational numbers.