| L(s) = 1 | + (0.968 − 1.43i)3-s + (−2.72 − 0.993i)5-s + (0.186 − 1.06i)7-s + (−1.12 − 2.78i)9-s + (3.28 − 1.19i)11-s + (−3.77 + 3.17i)13-s + (−4.07 + 2.95i)15-s + (−3.54 − 6.13i)17-s + (−1.08 + 1.88i)19-s + (−1.34 − 1.29i)21-s + (−1.11 − 6.30i)23-s + (2.63 + 2.20i)25-s + (−5.08 − 1.08i)27-s + (2.20 + 1.85i)29-s + (−0.481 − 2.72i)31-s + ⋯ |
| L(s) = 1 | + (0.559 − 0.828i)3-s + (−1.22 − 0.444i)5-s + (0.0706 − 0.400i)7-s + (−0.374 − 0.927i)9-s + (0.990 − 0.360i)11-s + (−1.04 + 0.879i)13-s + (−1.05 + 0.763i)15-s + (−0.858 − 1.48i)17-s + (−0.249 + 0.431i)19-s + (−0.292 − 0.282i)21-s + (−0.231 − 1.31i)23-s + (0.526 + 0.441i)25-s + (−0.977 − 0.208i)27-s + (0.409 + 0.343i)29-s + (−0.0864 − 0.490i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.407675 - 0.993464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.407675 - 0.993464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.968 + 1.43i)T \) |
| good | 5 | \( 1 + (2.72 + 0.993i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.186 + 1.06i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.28 + 1.19i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.77 - 3.17i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.54 + 6.13i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.08 - 1.88i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.11 + 6.30i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.20 - 1.85i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.481 + 2.72i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.40 - 7.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.953 - 0.800i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.05 + 2.56i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.575 + 3.26i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 1.88T + 53T^{2} \) |
| 59 | \( 1 + (-13.0 - 4.75i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.65 + 15.0i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.47 + 1.23i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.650 + 1.12i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.86 + 4.95i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.35 - 6.16i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.95 + 1.64i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.77 + 3.07i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.41 - 2.33i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.16681647616712272570542232687, −9.626728742986035114007139206623, −8.832273998805506806077477281045, −8.048897995894788828482167413875, −7.12877977616385858937772873424, −6.52752487570325085086901706605, −4.67574039582982306436357312122, −3.88477447164457965853102081865, −2.43015482804395367051519814059, −0.63527818393621439652986073972,
2.42242659259311461668851926173, 3.71536785523240611817107793613, 4.34167236940148316179776037384, 5.66775398989691112295864896206, 7.10579595564319622660799697775, 7.922794033055376642152634957998, 8.763940548810914711323158448784, 9.660263383454832533024846134832, 10.63650697422973891521633754169, 11.36189061477385364128599814779
