| L(s) = 1 | + (4.49 + 0.792i)2-s + (8.84 + 1.64i)3-s + (4.54 + 1.65i)4-s + (−3.40 − 4.05i)5-s + (38.4 + 14.4i)6-s + (−28.7 + 10.4i)7-s + (−44.1 − 25.4i)8-s + (75.5 + 29.1i)9-s + (−12.0 − 20.9i)10-s + (13.9 − 16.6i)11-s + (37.4 + 22.1i)12-s + (0.515 + 2.92i)13-s + (−137. + 24.2i)14-s + (−23.4 − 41.5i)15-s + (−237. − 199. i)16-s + (−249. + 144. i)17-s + ⋯ |
| L(s) = 1 | + (1.12 + 0.198i)2-s + (0.983 + 0.183i)3-s + (0.283 + 0.103i)4-s + (−0.136 − 0.162i)5-s + (1.06 + 0.400i)6-s + (−0.586 + 0.213i)7-s + (−0.689 − 0.398i)8-s + (0.932 + 0.360i)9-s + (−0.120 − 0.209i)10-s + (0.115 − 0.137i)11-s + (0.260 + 0.153i)12-s + (0.00305 + 0.0172i)13-s + (−0.700 + 0.123i)14-s + (−0.104 − 0.184i)15-s + (−0.927 − 0.778i)16-s + (−0.864 + 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.50825 + 0.329975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.50825 + 0.329975i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.84 - 1.64i)T \) |
| good | 2 | \( 1 + (-4.49 - 0.792i)T + (15.0 + 5.47i)T^{2} \) |
| 5 | \( 1 + (3.40 + 4.05i)T + (-108. + 615. i)T^{2} \) |
| 7 | \( 1 + (28.7 - 10.4i)T + (1.83e3 - 1.54e3i)T^{2} \) |
| 11 | \( 1 + (-13.9 + 16.6i)T + (-2.54e3 - 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-0.515 - 2.92i)T + (-2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (249. - 144. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-220. + 382. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (301. - 828. i)T + (-2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-1.06e3 - 187. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-1.63e3 - 595. i)T + (7.07e5 + 5.93e5i)T^{2} \) |
| 37 | \( 1 + (-293. - 508. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.37e3 - 242. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.22e3 + 1.03e3i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (754. + 2.07e3i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 - 1.53e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (844. + 1.00e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-2.81e3 + 1.02e3i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (1.42e3 + 8.05e3i)T + (-1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (4.74e3 - 2.73e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (1.14e3 - 1.98e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.00e3 + 5.71e3i)T + (-3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (5.85e3 + 1.03e3i)T + (4.45e7 + 1.62e7i)T^{2} \) |
| 89 | \( 1 + (-1.84e3 - 1.06e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (3.15e3 + 2.64e3i)T + (1.53e7 + 8.71e7i)T^{2} \) |
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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
−15.84592349827681567410369870167, −15.36892852857331841771609417557, −13.93122996100013213365534314110, −13.32124853148448932255665162819, −12.01420199515637217839142989513, −9.860631349562525144425129638473, −8.593521352740820663466114338139, −6.61814081854728613707392465927, −4.66438775164978108650333465686, −3.13713605388541925588825539003,
2.87740708847819798890813687801, 4.33280365031461167121644758835, 6.56313039698144668963889800541, 8.390333856508080783657643237088, 9.907908998032771019150315116747, 11.91882973043762537546604437970, 13.04381349653035326490502274178, 13.92589445621677501123915909146, 14.87233072313932085911755580746, 16.04895659217528744367295552033
