The factor 35/34: fourth powers in the mean square of L(1, χ) over quartic characters

Restricted to the odd quartic characters modulo primes p ≡ 5 (mod 8), Louboutin’s mean square of L(1, χ) with the Euler factor at 2 removed has limit (35/34)·π²/8 instead of π²/8. We explain the factor and prove it, together with its analogue for characters of any order r with the Euler factors at any finite set N of primes removed, within the random Euler product model. The order-r local factor is (1 + q^(−r))/((1 − q^(−2))(1 − q^(−r))). The excess over the full family is the product over q ∉ N of (1 + q^(−r))/(1 − q^(−r)), which equals ζ^(N)(r)²/ζ^(N)(2r). It is also the sum of (ab)^(−r) over the coprime pairs of integers prime to N, the weight of the pairs (m, n) with m/n an r-th power. For even r this factor is rational; it exceeds 1, decreases strictly in r, and tends to 1. For r = 4 and N = {2} it is 35/34. With the factor 4/5 that the congruence forces at 2, it gives 7π²/68, so the mean 7/136 of h⁻/p for the imaginary cyclic quartic fields of prime conductor is (1/20)·(35/34). Every statement about the model and the constants is formally verified in Lean 4 with Mathlib. Version 2 adds Proposition 3.3 and Section 5 (orthogonality for characters of order dividing r at a single prime) and separates the two full-family baselines π²/8 and π²/10.