| L(s) = 1 | + (0.939 + 0.342i)2-s + (1.28 + 1.16i)3-s + (0.766 + 0.642i)4-s + (−0.345 + 1.96i)5-s + (0.809 + 1.53i)6-s + (0.766 − 0.642i)7-s + (0.500 + 0.866i)8-s + (0.298 + 2.98i)9-s + (−0.995 + 1.72i)10-s + (−0.732 − 4.15i)11-s + (0.236 + 1.71i)12-s + (−2.94 + 1.07i)13-s + (0.939 − 0.342i)14-s + (−2.72 + 2.11i)15-s + (0.173 + 0.984i)16-s + (2.40 − 4.17i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.741 + 0.670i)3-s + (0.383 + 0.321i)4-s + (−0.154 + 0.876i)5-s + (0.330 + 0.625i)6-s + (0.289 − 0.242i)7-s + (0.176 + 0.306i)8-s + (0.0996 + 0.995i)9-s + (−0.314 + 0.545i)10-s + (−0.220 − 1.25i)11-s + (0.0683 + 0.495i)12-s + (−0.817 + 0.297i)13-s + (0.251 − 0.0914i)14-s + (−0.703 + 0.546i)15-s + (0.0434 + 0.246i)16-s + (0.583 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.87362 + 1.48790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87362 + 1.48790i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.28 - 1.16i)T \) |
| 7 | \( 1 + (-0.766 + 0.642i)T \) |
| good | 5 | \( 1 + (0.345 - 1.96i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.732 + 4.15i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.94 - 1.07i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.40 + 4.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 3.32i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.32 + 6.14i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.42 - 2.33i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.95 - 1.63i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.96 + 3.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.70 - 0.986i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.641 + 3.63i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.79 + 7.37i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + (0.543 - 3.08i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.30 - 2.77i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-13.4 + 4.89i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.02 - 5.24i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.22 - 7.32i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (13.3 + 4.85i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (3.32 + 1.20i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.49 + 9.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.904 - 5.12i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.53008763103379031509417043052, −10.57531901547476843180805179694, −9.965496025546221999448756509121, −8.592342294350998183913221581464, −7.81410421131942621629964354504, −6.88086625150252026389846963664, −5.60736911639104194041070520876, −4.51490011662368178516876789833, −3.39245059769572752572226815377, −2.57163750038956626740334843221,
1.50231397100992177844936761995, 2.70088362405388875132879350850, 4.16658515653918513342486496320, 5.09951115081013227613944673577, 6.34017647476508043384081177944, 7.60514286807216958353868387475, 8.126931909338688254807701637645, 9.432187127769188518990624980489, 10.06996673660444358610095818861, 11.63164856937493903082024570068
