| L(s) = 1 | + (0.939 − 0.342i)2-s + (−1.01 + 1.40i)3-s + (0.766 − 0.642i)4-s + (2.27 − 1.91i)5-s + (−0.470 + 1.66i)6-s + (−0.625 − 2.57i)7-s + (0.500 − 0.866i)8-s + (−0.952 − 2.84i)9-s + (1.48 − 2.57i)10-s + (−1.31 − 1.10i)11-s + (0.128 + 1.72i)12-s + (1.50 − 1.26i)13-s + (−1.46 − 2.20i)14-s + (0.382 + 5.13i)15-s + (0.173 − 0.984i)16-s + (−0.454 + 0.787i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.584 + 0.811i)3-s + (0.383 − 0.321i)4-s + (1.01 − 0.855i)5-s + (−0.191 + 0.680i)6-s + (−0.236 − 0.971i)7-s + (0.176 − 0.306i)8-s + (−0.317 − 0.948i)9-s + (0.470 − 0.814i)10-s + (−0.395 − 0.331i)11-s + (0.0370 + 0.498i)12-s + (0.417 − 0.349i)13-s + (−0.392 − 0.588i)14-s + (0.0987 + 1.32i)15-s + (0.0434 − 0.246i)16-s + (−0.110 + 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.71752 - 0.681051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71752 - 0.681051i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.01 - 1.40i)T \) |
| 7 | \( 1 + (0.625 + 2.57i)T \) |
| good | 5 | \( 1 + (-2.27 + 1.91i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (1.31 + 1.10i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.50 + 1.26i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.454 - 0.787i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.11 - 5.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.70 - 1.71i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.68 + 2.25i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.75 - 2.30i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + (-7.48 + 6.28i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.63 - 2.41i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.46 - 5.42i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-2.29 - 3.96i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 10.2i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.01 - 3.36i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (9.74 + 3.54i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.48 - 2.57i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 + (12.0 - 4.40i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (11.2 + 9.42i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.42 + 7.65i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.456 - 0.166i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.09962315397620688674591853585, −10.41383724229218209374168543025, −9.726189299307464648395920199110, −8.787528044608688335950762557195, −7.27095535175923504497350922549, −5.87235616216702469188667557892, −5.48329647333903085501471434602, −4.33023396309840159862343562442, −3.30557621882957932900208567092, −1.22004734614314424376703349367,
2.07252727870201347458391362493, 2.94910552756637413393781813288, 5.04197928347970890955823484333, 5.72110688572466467511787455794, 6.66703120887136823743725455034, 7.18230597834221749725258766077, 8.626919940358448732206862544085, 9.714976513879302977911052722553, 10.92595050726810659582174480882, 11.48149138067747454795167490349
