| L(s) = 1 | + (−1.12 − 6.39i)2-s + (−39.6 − 24.8i)3-s + (80.7 − 29.3i)4-s + (−233. − 196. i)5-s + (−114. + 281. i)6-s + (100. + 36.6i)7-s + (−693. − 1.20e3i)8-s + (953. + 1.96e3i)9-s + (−989. + 1.71e3i)10-s + (−3.52e3 + 2.95e3i)11-s + (−3.92e3 − 840. i)12-s + (−1.36e3 + 7.72e3i)13-s + (120. − 685. i)14-s + (4.39e3 + 1.35e4i)15-s + (1.52e3 − 1.27e3i)16-s + (−4.54e3 + 7.86e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0995 − 0.564i)2-s + (−0.847 − 0.530i)3-s + (0.630 − 0.229i)4-s + (−0.836 − 0.701i)5-s + (−0.215 + 0.531i)6-s + (0.111 + 0.0404i)7-s + (−0.479 − 0.830i)8-s + (0.436 + 0.899i)9-s + (−0.313 + 0.542i)10-s + (−0.798 + 0.669i)11-s + (−0.656 − 0.140i)12-s + (−0.171 + 0.974i)13-s + (0.0117 − 0.0667i)14-s + (0.336 + 1.03i)15-s + (0.0929 − 0.0780i)16-s + (−0.224 + 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.149507 + 0.300651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149507 + 0.300651i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (39.6 + 24.8i)T \) |
| good | 2 | \( 1 + (1.12 + 6.39i)T + (-120. + 43.7i)T^{2} \) |
| 5 | \( 1 + (233. + 196. i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-100. - 36.6i)T + (6.30e5 + 5.29e5i)T^{2} \) |
| 11 | \( 1 + (3.52e3 - 2.95e3i)T + (3.38e6 - 1.91e7i)T^{2} \) |
| 13 | \( 1 + (1.36e3 - 7.72e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (4.54e3 - 7.86e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.06e4 + 1.83e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.45e4 - 1.25e4i)T + (2.60e9 - 2.18e9i)T^{2} \) |
| 29 | \( 1 + (3.74e4 + 2.12e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-1.30e5 + 4.75e4i)T + (2.10e10 - 1.76e10i)T^{2} \) |
| 37 | \( 1 + (1.85e5 - 3.21e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-4.55e4 + 2.58e5i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (6.87e5 - 5.76e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (-6.44e3 - 2.34e3i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 + 1.05e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.49e5 + 2.92e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-1.82e6 - 6.62e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-7.14e5 + 4.05e6i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (8.90e5 - 1.54e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.00e6 - 1.74e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.46e6 + 8.33e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (1.43e6 + 8.12e6i)T + (-2.54e13 + 9.28e12i)T^{2} \) |
| 89 | \( 1 + (-5.36e6 - 9.29e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-3.70e6 + 3.10e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−15.33964921331303696583374234864, −13.17662643231699911350049611785, −12.05197644207921095438702669421, −11.42314280595265911455785526285, −10.01861277845411067202661071764, −7.916317212715942319098496462074, −6.49871698295879328630049861549, −4.61552586460469134744286529860, −1.93526730419328961957377591351, −0.18096204803092835100479506295,
3.28497586871587180483334415050, 5.46272861724934231714355825308, 6.90144912102463713715853596563, 8.169035852576008960849981666576, 10.48738589634382077860462290070, 11.25884059696581134261670802448, 12.45420439504393798185912471945, 14.65385776292221495858692312959, 15.64431945053019223757038042554, 16.24647198324022341244657670388
