| L(s) = 1 | + (0.342 − 0.939i)2-s + (−0.811 − 1.53i)3-s + (−0.766 − 0.642i)4-s + (0.307 − 1.74i)5-s + (−1.71 + 0.238i)6-s + (2.14 − 1.54i)7-s + (−0.866 + 0.500i)8-s + (−1.68 + 2.48i)9-s + (−1.53 − 0.886i)10-s + (2.04 − 0.360i)11-s + (−0.362 + 1.69i)12-s + (−1.92 − 5.30i)13-s + (−0.722 − 2.54i)14-s + (−2.92 + 0.945i)15-s + (0.173 + 0.984i)16-s + (−1.73 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.468 − 0.883i)3-s + (−0.383 − 0.321i)4-s + (0.137 − 0.781i)5-s + (−0.700 + 0.0975i)6-s + (0.810 − 0.585i)7-s + (−0.306 + 0.176i)8-s + (−0.561 + 0.827i)9-s + (−0.485 − 0.280i)10-s + (0.616 − 0.108i)11-s + (−0.104 + 0.488i)12-s + (−0.535 − 1.47i)13-s + (−0.193 − 0.680i)14-s + (−0.754 + 0.244i)15-s + (0.0434 + 0.246i)16-s + (−0.420 + 0.729i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.141478 - 1.24509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.141478 - 1.24509i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 3 | \( 1 + (0.811 + 1.53i)T \) |
| 7 | \( 1 + (-2.14 + 1.54i)T \) |
| good | 5 | \( 1 + (-0.307 + 1.74i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.04 + 0.360i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.92 + 5.30i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.73 - 3.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.552 - 0.318i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.38 - 5.22i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.536 - 1.47i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.11 - 3.71i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.87 + 6.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-11.6 + 4.22i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.03 + 11.5i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.88 - 1.57i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 4.08iT - 53T^{2} \) |
| 59 | \( 1 + (0.784 - 4.44i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.59 - 9.04i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 0.943i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.13 + 3.54i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.60 + 4.39i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.9 - 5.09i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.01 + 0.731i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.14 + 3.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.7 + 2.77i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.00316280328613934566934865076, −10.41080569064675788935078918594, −9.056640262748102320721541509781, −8.114359131479145912638828845302, −7.27459985005790164434836353863, −5.79815992163325878231187887332, −5.13002945352343307812113708953, −3.85691889767510436057731164633, −2.04172262564576225471627654103, −0.868124952456304899024357835330,
2.54123607017069290542992401074, 4.22259802534510943004237495867, 4.81033162165621536573910963570, 6.16513228840607423939211211314, 6.71895046601563063289198214840, 8.060132928489950744392105523258, 9.190725203740400391974451115451, 9.742266261844123497690625079938, 11.12785148645478526644382595555, 11.54420656078682483350289886842
