| L(s) = 1 | + (16.7 − 6.11i)2-s + (46.7 + 1.15i)3-s + (146. − 122. i)4-s + (−21.3 − 121. i)5-s + (792. − 266. i)6-s + (−285. − 239. i)7-s + (565. − 980. i)8-s + (2.18e3 + 107. i)9-s + (−1.10e3 − 1.90e3i)10-s + (−803. + 4.55e3i)11-s + (6.99e3 − 5.58e3i)12-s + (933. + 339. i)13-s + (−6.26e3 − 2.27e3i)14-s + (−860. − 5.69e3i)15-s + (−740. + 4.20e3i)16-s + (−1.11e4 − 1.93e4i)17-s + ⋯ |
| L(s) = 1 | + (1.48 − 0.540i)2-s + (0.999 + 0.0246i)3-s + (1.14 − 0.960i)4-s + (−0.0765 − 0.434i)5-s + (1.49 − 0.503i)6-s + (−0.314 − 0.264i)7-s + (0.390 − 0.676i)8-s + (0.998 + 0.0493i)9-s + (−0.348 − 0.602i)10-s + (−0.182 + 1.03i)11-s + (1.16 − 0.932i)12-s + (0.117 + 0.0428i)13-s + (−0.609 − 0.221i)14-s + (−0.0657 − 0.435i)15-s + (−0.0452 + 0.256i)16-s + (−0.551 − 0.954i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.16168 - 1.83444i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.16168 - 1.83444i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-46.7 - 1.15i)T \) |
| good | 2 | \( 1 + (-16.7 + 6.11i)T + (98.0 - 82.2i)T^{2} \) |
| 5 | \( 1 + (21.3 + 121. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (285. + 239. i)T + (1.43e5 + 8.11e5i)T^{2} \) |
| 11 | \( 1 + (803. - 4.55e3i)T + (-1.83e7 - 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-933. - 339. i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.11e4 + 1.93e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.38e4 - 2.40e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.33e4 + 1.95e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.46e5 - 5.33e4i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (2.45e5 - 2.06e5i)T + (4.77e9 - 2.70e10i)T^{2} \) |
| 37 | \( 1 + (8.81e4 + 1.52e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-4.56e5 - 1.66e5i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-1.09e5 + 6.19e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (7.84e4 + 6.58e4i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 - 2.07e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (4.42e5 + 2.50e6i)T + (-2.33e12 + 8.51e11i)T^{2} \) |
| 61 | \( 1 + (1.00e6 + 8.40e5i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (2.45e6 + 8.93e5i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-4.29e5 - 7.44e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.52e6 + 2.64e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (6.42e5 - 2.33e5i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-1.46e6 + 5.35e5i)T + (2.07e13 - 1.74e13i)T^{2} \) |
| 89 | \( 1 + (1.37e6 - 2.37e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.45e5 + 8.26e5i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−15.06690197187789764474324189667, −14.21175962206584853244343222014, −13.03537712033033962574315688659, −12.40552005408849694359944326463, −10.61597209672876433992127259858, −9.013330030804004113969920547073, −7.08089892400597355958727807096, −4.92298174544735185252840111452, −3.65607764552942645942855083294, −2.06757174052282325825117870834,
2.77697507314898640161983936999, 4.01675366444300590245605743438, 5.95693250709098969585533285034, 7.35135119755347123621473730370, 9.008853407441475344111587443076, 11.05650035963131398495521214870, 12.88635844697360454148219730108, 13.45744595197499751759356099475, 14.76466525090176963780133654133, 15.30132677129067448826941379071
