| 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
−11.63164856937493903082024570068, −10.06996673660444358610095818861, −9.432187127769188518990624980489, −8.126931909338688254807701637645, −7.60514286807216958353868387475, −6.34017647476508043384081177944, −5.09951115081013227613944673577, −4.16658515653918513342486496320, −2.70088362405388875132879350850, −1.50231397100992177844936761995,
2.57163750038956626740334843221, 3.39245059769572752572226815377, 4.51490011662368178516876789833, 5.60736911639104194041070520876, 6.88086625150252026389846963664, 7.81410421131942621629964354504, 8.592342294350998183913221581464, 9.965496025546221999448756509121, 10.57531901547476843180805179694, 11.53008763103379031509417043052
