| L(s) = 1 | + 0.536i·2-s − i·3-s + 1.71·4-s + 0.536·6-s + 2.57i·7-s + 1.99i·8-s − 9-s − 5.14·11-s − 1.71i·12-s + 1.47i·13-s − 1.38·14-s + 2.35·16-s + 0.687i·17-s − 0.536i·18-s + 8.09·19-s + ⋯ |
| L(s) = 1 | + 0.379i·2-s − 0.577i·3-s + 0.856·4-s + 0.219·6-s + 0.972i·7-s + 0.704i·8-s − 0.333·9-s − 1.55·11-s − 0.494i·12-s + 0.409i·13-s − 0.368·14-s + 0.588·16-s + 0.166i·17-s − 0.126i·18-s + 1.85·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.757020532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757020532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.536iT - 2T^{2} \) |
| 7 | \( 1 - 2.57iT - 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 - 1.47iT - 13T^{2} \) |
| 17 | \( 1 - 0.687iT - 17T^{2} \) |
| 19 | \( 1 - 8.09T + 19T^{2} \) |
| 23 | \( 1 + 0.372iT - 23T^{2} \) |
| 29 | \( 1 + 0.0356T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 - 1.90iT - 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 8.52iT - 47T^{2} \) |
| 53 | \( 1 - 9.12iT - 53T^{2} \) |
| 59 | \( 1 - 0.176T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 - 0.0834iT - 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 4.95T + 79T^{2} \) |
| 83 | \( 1 + 9.36iT - 83T^{2} \) |
| 89 | \( 1 - 0.0123T + 89T^{2} \) |
| 97 | \( 1 + 7.62iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.321454643709080481084812282733, −8.378999914160104738786831116604, −7.69755356172679418838914506424, −7.23007322852048175772026819695, −6.17146296967516583075317574550, −5.60943258617839349848584696430, −4.91659415124183777068758648332, −3.11345155703045996942736436609, −2.58389646702969636098140152724, −1.49886022040442335771160485359,
0.60969628027115919443968825257, 2.10302258569434093205369963045, 3.18794621301568533676371771586, 3.71653884986924876357979866903, 5.11808470565630478282608610967, 5.54290994423802529083328168023, 6.86313736957999975190972906587, 7.47327163144598043087873613797, 8.059231639996512188732327888217, 9.276410952393817326859317262963
