| L(s) = 1 | + (−0.907 − 2.79i)2-s + (21.8 + 15.8i)3-s + (96.5 − 70.1i)4-s + (61.3 − 188. i)5-s + (24.5 − 75.4i)6-s + (−198. + 144. i)7-s + (−587. − 427. i)8-s + (225. + 693. i)9-s − 583.·10-s + (3.84e3 − 2.16e3i)11-s + 3.22e3·12-s + (−840. − 2.58e3i)13-s + (582. + 423. i)14-s + (4.33e3 − 3.14e3i)15-s + (4.06e3 − 1.25e4i)16-s + (6.46e3 − 1.99e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0802 − 0.246i)2-s + (0.467 + 0.339i)3-s + (0.754 − 0.548i)4-s + (0.219 − 0.675i)5-s + (0.0463 − 0.142i)6-s + (−0.218 + 0.158i)7-s + (−0.405 − 0.294i)8-s + (0.103 + 0.317i)9-s − 0.184·10-s + (0.871 − 0.491i)11-s + 0.538·12-s + (−0.106 − 0.326i)13-s + (0.0567 + 0.0412i)14-s + (0.331 − 0.240i)15-s + (0.247 − 0.762i)16-s + (0.319 − 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.97060 - 1.21234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97060 - 1.21234i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-21.8 - 15.8i)T \) |
| 11 | \( 1 + (-3.84e3 + 2.16e3i)T \) |
| good | 2 | \( 1 + (0.907 + 2.79i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (-61.3 + 188. i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (198. - 144. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (840. + 2.58e3i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-6.46e3 + 1.99e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (1.17e4 + 8.51e3i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 - 5.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (2.38e3 - 1.73e3i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-4.77e4 - 1.47e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (3.94e5 - 2.86e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-1.41e5 - 1.02e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 7.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.21e5 - 3.78e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-7.04e4 - 2.16e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.31e6 + 9.53e5i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (3.26e5 - 1.00e6i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 - 1.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-3.28e5 + 1.01e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (1.12e6 - 8.15e5i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-2.00e6 - 6.17e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (2.62e6 - 8.07e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + 1.15e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.30e5 - 1.63e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.03676333800727050058430155740, −13.86619839673899957721723257936, −12.39567106021829267525793690055, −11.13869937903644663941380214921, −9.784247499149712850889270613373, −8.755851963829766376116309623801, −6.81509340957274778424732264190, −5.17420588517830959289036257549, −3.01920420674782536562027897069, −1.16116000401680487181239630478,
2.01328534309312186499163629957, 3.59348648241756034672466326249, 6.38113988690196384103033686663, 7.24571814566998405330249725507, 8.705634218983044143271720570071, 10.35552321552543947800765967499, 11.77978878131044656516620941534, 12.90438687343214452527793194178, 14.43605268245605630464930724876, 15.18390281793766677019095894169
