| L(s) = 1 | + (0.723 + 0.690i)2-s + (−0.116 − 0.810i)3-s + (0.0475 + 0.998i)4-s + (1.65 − 3.63i)5-s + (0.475 − 0.667i)6-s + (−0.754 + 3.10i)7-s + (−0.654 + 0.755i)8-s + (2.23 − 0.656i)9-s + (3.70 − 1.48i)10-s + (−1.84 − 2.59i)11-s + (0.804 − 0.155i)12-s + (−1.96 + 5.68i)13-s + (−2.69 + 1.72i)14-s + (−3.14 − 0.922i)15-s + (−0.995 + 0.0950i)16-s + (−0.0319 + 0.669i)17-s + ⋯ |
| L(s) = 1 | + (0.511 + 0.487i)2-s + (−0.0673 − 0.468i)3-s + (0.0237 + 0.499i)4-s + (0.742 − 1.62i)5-s + (0.194 − 0.272i)6-s + (−0.285 + 1.17i)7-s + (−0.231 + 0.267i)8-s + (0.744 − 0.218i)9-s + (1.17 − 0.469i)10-s + (−0.556 − 0.780i)11-s + (0.232 − 0.0447i)12-s + (−0.545 + 1.57i)13-s + (−0.719 + 0.462i)14-s + (−0.811 − 0.238i)15-s + (−0.248 + 0.0237i)16-s + (−0.00773 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44155 - 0.0185476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44155 - 0.0185476i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.723 - 0.690i)T \) |
| 67 | \( 1 + (-4.54 + 6.80i)T \) |
| good | 3 | \( 1 + (0.116 + 0.810i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-1.65 + 3.63i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (0.754 - 3.10i)T + (-6.22 - 3.20i)T^{2} \) |
| 11 | \( 1 + (1.84 + 2.59i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (1.96 - 5.68i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (0.0319 - 0.669i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.0715 - 0.294i)T + (-16.8 + 8.70i)T^{2} \) |
| 23 | \( 1 + (5.86 - 4.61i)T + (5.42 - 22.3i)T^{2} \) |
| 29 | \( 1 + (-3.22 + 5.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.152 - 0.440i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (0.160 + 0.277i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.45 - 4.35i)T + (23.7 - 33.3i)T^{2} \) |
| 43 | \( 1 + (-1.96 - 1.26i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-8.40 - 3.36i)T + (34.0 + 32.4i)T^{2} \) |
| 53 | \( 1 + (-7.31 + 4.69i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (5.95 - 6.87i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.05 - 1.48i)T + (-19.9 - 57.6i)T^{2} \) |
| 71 | \( 1 + (0.449 + 9.43i)T + (-70.6 + 6.74i)T^{2} \) |
| 73 | \( 1 + (-1.77 + 2.49i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-4.05 + 0.780i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-8.68 + 0.829i)T + (81.5 - 15.7i)T^{2} \) |
| 89 | \( 1 + (0.194 - 1.35i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.22 + 5.59i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.38491826276099282461263844897, −12.19369036094048819777353574304, −12.00633386791968032969098473780, −9.735852409999922601377377400350, −8.989754333753816036412776167991, −7.965393637521427784008932773238, −6.35707079392754094996068862817, −5.54810073607729317998094590713, −4.37607852470855079213296288325, −1.99940283490841713799065731810,
2.50574010285677487568648290912, 3.80129377904261842197972156224, 5.25390491834562060215122613487, 6.73400747839260942936459975638, 7.52090576874605163049502657748, 9.910582745988528155305365675052, 10.42865012188734110866228929283, 10.59758894915446500579720350268, 12.38715312753959319710533887354, 13.35864718811986480996689189228
