# SageMath with numberdb: analytic n=0 or 1, integer ell>=2.
# Returns an exact rational or an Arb enclosure with the requested significant digits.
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.rings.complex_arb import ComplexBallField

def love_analytic(n, ell, digits=100):
    n, ell = QQ(n), QQ(ell)
    if n not in (0, 1) or ell.denominator() != 1 or ell < 2:
        raise ValueError('Use n=0 or 1 and an integer ell>=2.')
    ell = ZZ(ell)
    if n == 0:
        return QQ(3) / (4 * (ell - 1))
    target = numberdb.bits(digits, losing=8)
    precision = target + 80
    while True:
        p = ComplexBallField(precision).pi()
        k = (p.bessel_J(ell + QQ(3)/2)
             / (2 * p.bessel_J(ell - QQ(1)/2))).real()
        if k.is_finite() and k.accuracy() >= target:
            return k
        precision *= 2

print(love_analytic(0, 4))
print(love_analytic(1, 4))