Computational Evidence for Logarithmic Scaling in Quadratic L-Function Extreme Values

We present a computational framework for investigating extreme value behavior in quadratic Dirichlet L-functions L(s, χp) on the critical line Re(s) = 1/2, where χp is the Legendre symbol modulo an odd prime p. Through high-precision numerical computation using 100-bit arithmetic, we introduce an original extreme value parameter E(p) and analyze its behavior for 99 primes in the range 3 ≤ p ≤ 541. Our computational methodology represents the first systematic framework for quantifying extreme values in L-function families at the individual conductor level. This addresses a significant gap in the literature where previous approaches focus on asymptotic bounds rather than specific numerical quantification. The main empirical discovery is a strong logarithmic relationship between E(p) and log(log(p)), characterized by: E(p) = 0.2020(±0.0081) · log(log(p)) + 0.5494(±0.0128) (1) This pattern exhibits exceptional statistical correlation (r = 0.9650, R2 = 0.9312) and explains 93.12% of the observed variance. The empirical coefficient α = 0.2020 is numerically consistent with theoretical expectations from Random Matrix Theory for families with orthogonal symmetry, differing by only 1.0% from the predicted value α ≈ 0.200. We extend our analysis to include systematic computations for the Riemann zeta function ζ(1/2 + it) using identical methodology, providing methodological validation and preliminary evidence for potential universality patterns across L-function families. These computational results provide the first systematic empirical evidence for logarithmic conductor dependence in L-function extreme values and establish a novel, generalizable framework for quantitative extreme value analysis across L-function families.