import mpmath as mp
with mp.workdps(130):
g1 = mp.findroot(mp.scorergi, (-1, 0))
print(mp.nstr(g1, 100))
The values are written to 100 significant digits. The generator uses Arb balls at 700 bits and a 900-term Maclaurin polynomial fixed by (1) and (2). Its coefficients satisfy $c_2=-1/(2\pi)$ and $c_{k+3}=c_k/((k+3)(k+2))$. Each residue class modulo three has a separate geometric bound for its omitted tail. The bounds for the function and its first two derivatives on $|x|\leq16$ are respectively less than $7\cdot10^{-432}$, $4\cdot10^{-430}$ and $3\cdot10^{-428}$; they are added to the ball radii.
The generator certifies exactly twelve roots on $[-227/16,0]$. It covers this interval by 227 adjacent segments of length $1/16$. Local function and derivative enclosures follow from the differential equation and a Gronwall bound. On 206 segments the function excludes zero. On the remaining 21 segments the derivative has a fixed nonzero sign: nine have equal endpoint signs and twelve have opposite endpoint signs. Thus each of the twelve latter segments contains exactly one root and no root is omitted. Within each such segment, an mpmath candidate is enclosed by exact rational endpoints of separation $2\cdot10^{-120}$. Ball evaluations certify opposite function signs at these endpoints and a nonzero derivative throughout the bracket. Each bracket lies strictly inside the interval represented by its stored decimal.
Independent certified quadrature of the Airy representation [5], using $\int_0^\infty\operatorname{Ai}(t)\,dt=1/3$ [6], also establishes the endpoint signs for every root. The quadrature controls integrate $t^2$ and the derivative of the Airy function, with known nonzero answers. The attached checks compare every root with mpmath [7] at 150 and 200 decimal digits, test the initial values, differential equation and sum identity, and repeat the endpoint checks on the intervals read from the database. The mpmath comparisons are numerical checks separate from the rigorous certificates.