| L(s) = 1 | + (25.1 − 30.0i)2-s + (−210. + 121. i)3-s + (−88.7 − 503. i)4-s + (−969. + 2.66e3i)5-s + (−1.63e3 + 9.37e3i)6-s + (2.75e3 − 1.56e4i)7-s + (1.74e4 + 1.00e4i)8-s + (2.93e4 − 5.12e4i)9-s + (5.55e4 + 9.61e4i)10-s + (−8.89e4 − 2.44e5i)11-s + (7.99e4 + 9.50e4i)12-s + (6.95e4 − 5.83e4i)13-s + (−3.99e5 − 4.76e5i)14-s + (−1.20e5 − 6.78e5i)15-s + (1.23e6 − 4.48e5i)16-s + (9.51e5 − 5.49e5i)17-s + ⋯ |
| L(s) = 1 | + (0.787 − 0.937i)2-s + (−0.865 + 0.501i)3-s + (−0.0866 − 0.491i)4-s + (−0.310 + 0.852i)5-s + (−0.210 + 1.20i)6-s + (0.163 − 0.929i)7-s + (0.531 + 0.306i)8-s + (0.497 − 0.867i)9-s + (0.555 + 0.961i)10-s + (−0.552 − 1.51i)11-s + (0.321 + 0.381i)12-s + (0.187 − 0.157i)13-s + (−0.743 − 0.885i)14-s + (−0.158 − 0.892i)15-s + (1.17 − 0.427i)16-s + (0.669 − 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.14953 - 1.44690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14953 - 1.44690i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (210. - 121. i)T \) |
| good | 2 | \( 1 + (-25.1 + 30.0i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (969. - 2.66e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (-2.75e3 + 1.56e4i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (8.89e4 + 2.44e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-6.95e4 + 5.83e4i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-9.51e5 + 5.49e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-3.29e5 + 5.71e5i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (5.95e6 - 1.05e6i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (-2.20e7 + 2.63e7i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (6.47e6 + 3.67e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (4.85e7 + 8.40e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-9.54e7 - 1.13e8i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-2.00e6 + 7.28e5i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (1.21e8 + 2.14e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 + 1.48e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (4.57e8 - 1.25e9i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (-8.02e6 + 4.54e7i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (-3.50e8 + 2.94e8i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (3.45e8 - 1.99e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-4.38e8 + 7.59e8i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (3.18e9 + 2.67e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (4.66e8 - 5.56e8i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (8.12e9 + 4.69e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-8.57e9 + 3.11e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−14.31573196097957262065832551560, −13.31479261941533085350678871143, −11.74443418630160992522320973687, −11.01361031087306275054390608487, −10.22965954450096427557911070806, −7.65709882479820317691699940958, −5.81706424671491614213506428553, −4.18099216220005249370029031196, −3.09810983081352207605168402219, −0.66024542569515764841140738133,
1.52531260612953286816156575820, 4.66532455428397263288197775348, 5.44054026839208826993914399614, 6.84789044804019019794867397387, 8.182093733422115608863614789055, 10.28046536825354462725467703318, 12.33141903821701182380548521665, 12.54968484205371050908179794241, 14.21374981055267630342077047882, 15.58540719415743063981499429419
