| L(s) = 1 | + (1.37 + 1.04i)3-s + (−1.84 − 1.26i)5-s + (−2.63 + 0.269i)7-s + (0.801 + 2.89i)9-s − 3.01i·11-s + 4.39·13-s + (−1.20 − 3.67i)15-s + 2.71i·17-s + 8.23i·19-s + (−3.91 − 2.38i)21-s − 3.81·23-s + (1.77 + 4.67i)25-s + (−1.92 + 4.82i)27-s − 1.17i·29-s − 1.73i·31-s + ⋯ |
| L(s) = 1 | + (0.796 + 0.605i)3-s + (−0.823 − 0.567i)5-s + (−0.994 + 0.101i)7-s + (0.267 + 0.963i)9-s − 0.908i·11-s + 1.21·13-s + (−0.311 − 0.950i)15-s + 0.657i·17-s + 1.88i·19-s + (−0.853 − 0.521i)21-s − 0.796·23-s + (0.355 + 0.934i)25-s + (−0.370 + 0.928i)27-s − 0.217i·29-s − 0.311i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243017614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243017614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.37 - 1.04i)T \) |
| 5 | \( 1 + (1.84 + 1.26i)T \) |
| 7 | \( 1 + (2.63 - 0.269i)T \) |
| good | 11 | \( 1 + 3.01iT - 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 - 2.71iT - 17T^{2} \) |
| 19 | \( 1 - 8.23iT - 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 + 1.17iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.60iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 9.18iT - 43T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 - 13.5iT - 61T^{2} \) |
| 67 | \( 1 - 0.494iT - 67T^{2} \) |
| 71 | \( 1 - 5.15iT - 71T^{2} \) |
| 73 | \( 1 - 4.76T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 8.16iT - 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−9.523312128064764943460893520466, −8.622338921335358430278450195168, −8.294074872349899497860051559685, −7.54164877828896835304694617456, −6.18407492833855512532805733510, −5.65975945975817035231241482952, −4.18970881207313843122574071244, −3.75628556619813735864918017566, −3.04073586344036105793479405609, −1.43810827330623188246711903993,
0.43587683466064235955199389134, 2.10470598104000542610134236783, 3.15875134539586364472957450833, 3.69326770422918052421384739232, 4.79025228432364982554240008095, 6.27105695120521108258896328270, 6.97380570078108912663969505096, 7.24373619808132828791386467285, 8.403736422519885683473575780843, 8.910002238856583227413369865166
