Here is Hardy & Wright's answer from "An Introduction to the Theory of Numbers", (5th ed, p344), where they discuss a similar number:

"Although ... gives a 'formula' for the nth prime, it is not a very useful one. To calculate $p_n$ from this formula, it is necessary to know the value of $a$ correct to $2^n$ decimal places; and to do this, it is necessary to know the values of $p_1$, $p_2$, ..., $p_n$ ... There are a number of similar formulae which suffer from the same defect ... Any one of these formulae (or any similar one) would attain a different status if the exact value of the number $a$ which occurs in it could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."