| L(s) = 1 | + (−0.930 − 0.365i)2-s + (−0.0758 − 1.73i)3-s + (0.733 + 0.680i)4-s + (−2.77 − 1.89i)5-s + (−0.561 + 1.63i)6-s + (−2.14 + 1.54i)7-s + (−0.433 − 0.900i)8-s + (−2.98 + 0.262i)9-s + (1.89 + 2.77i)10-s + (0.905 + 6.00i)11-s + (1.12 − 1.32i)12-s + (1.32 − 1.05i)13-s + (2.56 − 0.654i)14-s + (−3.06 + 4.94i)15-s + (0.0747 + 0.997i)16-s + (−0.409 − 0.126i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 − 0.258i)2-s + (−0.0438 − 0.999i)3-s + (0.366 + 0.340i)4-s + (−1.24 − 0.845i)5-s + (−0.229 + 0.668i)6-s + (−0.811 + 0.584i)7-s + (−0.153 − 0.318i)8-s + (−0.996 + 0.0875i)9-s + (0.597 + 0.876i)10-s + (0.272 + 1.81i)11-s + (0.323 − 0.381i)12-s + (0.368 − 0.293i)13-s + (0.685 − 0.174i)14-s + (−0.790 + 1.27i)15-s + (0.0186 + 0.249i)16-s + (−0.0993 − 0.0306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0117487 + 0.0172140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0117487 + 0.0172140i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.930 + 0.365i)T \) |
| 3 | \( 1 + (0.0758 + 1.73i)T \) |
| 7 | \( 1 + (2.14 - 1.54i)T \) |
| good | 5 | \( 1 + (2.77 + 1.89i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.905 - 6.00i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 1.05i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.409 + 0.126i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.985 + 0.568i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.51 + 4.91i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (7.31 - 1.66i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.31 + 0.757i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.65 - 7.09i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (9.57 - 4.61i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (6.51 + 3.13i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.97 + 5.02i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (0.516 - 0.556i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (7.56 - 5.15i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (3.47 + 3.74i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.86 + 6.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.2 - 3.03i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (1.42 - 0.559i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (2.37 + 4.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.83 + 3.54i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.81 - 0.574i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 4.18iT - 97T^{2} \) |
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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
−12.26528411824243717030465250523, −11.45612666168808340721920010101, −10.08133075387657649237021637199, −8.998020287591067928581220165680, −8.300598139873502154975543737521, −7.34815201221961675003617173696, −6.57836011980463426679776374717, −4.98093301312818178153865369674, −3.46224866567673495702075719463, −1.84123487052234960793802802321,
0.01855470263861703250851748164, 3.41378309069636032261004374215, 3.69453475354283410941767010358, 5.64316236714076041051691402235, 6.66375163640761394731865847999, 7.73198353223053309667435484078, 8.691095930991834223908704469060, 9.585274821953043280062127311816, 10.72667526650137426590154410351, 11.10252850563848246108792181254
