| L(s) = 1 | − 3-s − 4.72i·7-s + 9-s + 3.93i·11-s − 3.46·13-s + 3.51i·17-s − 5.44i·19-s + 4.72i·21-s − 7.11i·23-s − 27-s + 3.66i·29-s − 5.23·31-s − 3.93i·33-s + 0.414·37-s + 3.46·39-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78i·7-s + 0.333·9-s + 1.18i·11-s − 0.961·13-s + 0.852i·17-s − 1.24i·19-s + 1.03i·21-s − 1.48i·23-s − 0.192·27-s + 0.681i·29-s − 0.940·31-s − 0.684i·33-s + 0.0681·37-s + 0.555·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04695268197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04695268197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.72iT - 7T^{2} \) |
| 11 | \( 1 - 3.93iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.51iT - 17T^{2} \) |
| 19 | \( 1 + 5.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 0.414T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 0.925iT - 47T^{2} \) |
| 53 | \( 1 + 0.233T + 53T^{2} \) |
| 59 | \( 1 - 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 0.563iT - 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 - 7.27iT - 97T^{2} \) |
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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
−8.451577271743216153975936254681, −7.34780205368592084686859388385, −7.13582263634538857749027962988, −6.42276791778489221267339331369, −5.12734631369575974160138354845, −4.50735452158809162414622533309, −3.91909144261263360741485436495, −2.52367211734074371282115686106, −1.27167987093273993867281557834, −0.01753273186891396842049831968,
1.71776983788306367631335939339, 2.74389657775264849351441669075, 3.64481632374148504579499884444, 5.00098180411852656944553697766, 5.59512481378970106817749919258, 5.98930079021340598180806715383, 7.05567354159986373570450648937, 7.976545174709070439219398854472, 8.615414494542148917598624963042, 9.579903481404833479860048148677
