Testing a formula puporting to give the k-th prime number

vengy.py

# testing a formula for the k-th prime number
# https://math.stackexchange.com/questions/5019919/an-exact-formula-for-the-k-th-prime-number

from mpmath import ceil, ln, mp, mpf, zeta

mp.dps = 500
mp.pretty = True

one = mpf("1")
two = mpf("2")
NPRIME = 30
primes = [one for i in range(NPRIME)]
primes[1] = mpf("2")
primes[2] = mpf("3")


def vengy_prime(k):
    s = 2 * k * ln(k * ln(k))
    prod = one
    for j in range(1, k):
        prod *= one - primes[j] ** (-s)
    # print(repr(prod))
    f = (1 - (zeta(s) * prod) ** (-1)) ** (-one / s)
    return ceil(f)


for k in range(3, NPRIME):
    primes[k] = vengy_prime(k)
print(primes)