| L(s) = 1 | − 6i·2-s + 28·4-s + 240i·5-s + 299·7-s − 552i·8-s + 1.44e3·10-s + 624i·11-s + 2.49e3·13-s − 1.79e3i·14-s − 1.52e3·16-s − 1.87e3i·17-s − 2.50e3·19-s + 6.72e3i·20-s + 3.74e3·22-s + 1.43e4i·23-s + ⋯ |
| L(s) = 1 | − 0.750i·2-s + 0.437·4-s + 1.91i·5-s + 0.871·7-s − 1.07i·8-s + 1.43·10-s + 0.468i·11-s + 1.13·13-s − 0.653i·14-s − 0.371·16-s − 0.381i·17-s − 0.365·19-s + 0.839i·20-s + 0.351·22-s + 1.17i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.96992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96992\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 6iT - 64T^{2} \) |
| 5 | \( 1 - 240iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 299T + 1.17e5T^{2} \) |
| 11 | \( 1 - 624iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.49e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.87e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.50e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.43e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.37e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.33e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 3.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 6.61e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 7.06e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.98e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.90e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.37e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 6.18e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.30e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.51e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.51e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.60e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.97e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.70e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.20e5T + 8.32e11T^{2} \) |
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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
−15.67967349470792618529683367992, −14.83552838579428360049450296384, −13.57293222126820040829193728746, −11.62369540682971623391568410023, −11.05625979805148337460201676372, −9.990486139180967140208432314978, −7.60224623730087719735163842154, −6.36916925635655843099364052175, −3.53858300520273510886731839187, −2.03232292531174936133167360743,
1.37340758254179577886894900599, 4.68841056120705700074402565526, 6.00095655174430841648749994050, 8.111968754919570520732082179980, 8.733367916535631938827728697853, 11.01828444618274095997147346912, 12.29168464788704310873779048123, 13.60520871549574784535648973829, 15.05649146261494812633409485461, 16.32901971440025001174604379715
