Lower bound of real primitive L-function at s=1

Sums of seven octahedral numbers

“…Then by assumption we have d = d ′ . From Lemma 2 of [7], for any 1 − ǫ < s < 1 such that L(s, χ −4d ) ≥ 0 and for any x ≥ 1 we have…”

“…Plugging in x = 10 4 (dd ′ ) 3 2 , and using the bound L(1, χ D ) ≤ log(4dd ′ )+1.44 2 from [7], we get…”

Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms

“…(N.B. The constants in Tatuzawa's result have been improved in [6] and [7]; these could be applied at the expense of slightly more complicated statements. )…”

On the Siegel–Tatuzawa–Hoffstein theorem

“…In this paper we follow the proof of Ji and Lu [6], with the following improvement: First, β and β 1 do not play the same roles in Ji and Lu [6], but they do in this paper (see (5) in the next section). Secondly, we use the following fact: the function xd −A 2 x F increases for 0 < x ≤ 1/(A 2 log d F ) and decreases for x ≥ 1/(A 2 log d F ).…”