| L(s) = 1 | + (−3.51 − 2.94i)2-s + (23.1 − 40.6i)3-s + (−18.5 − 105. i)4-s + (493. + 179. i)5-s + (−200. + 74.7i)6-s + (70.9 − 402. i)7-s + (−538. + 932. i)8-s + (−1.11e3 − 1.87e3i)9-s + (−1.20e3 − 2.08e3i)10-s + (2.18e3 − 795. i)11-s + (−4.71e3 − 1.67e3i)12-s + (1.99e3 − 1.67e3i)13-s + (−1.43e3 + 1.20e3i)14-s + (1.87e4 − 1.59e4i)15-s + (−8.22e3 + 2.99e3i)16-s + (−1.90e4 − 3.29e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.310 − 0.260i)2-s + (0.494 − 0.869i)3-s + (−0.145 − 0.823i)4-s + (1.76 + 0.643i)5-s + (−0.379 + 0.141i)6-s + (0.0781 − 0.443i)7-s + (−0.371 + 0.644i)8-s + (−0.511 − 0.859i)9-s + (−0.381 − 0.659i)10-s + (0.495 − 0.180i)11-s + (−0.787 − 0.280i)12-s + (0.251 − 0.211i)13-s + (−0.139 + 0.117i)14-s + (1.43 − 1.21i)15-s + (−0.502 + 0.182i)16-s + (−0.938 − 1.62i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.27245 - 1.57548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27245 - 1.57548i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-23.1 + 40.6i)T \) |
| good | 2 | \( 1 + (3.51 + 2.94i)T + (22.2 + 126. i)T^{2} \) |
| 5 | \( 1 + (-493. - 179. i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-70.9 + 402. i)T + (-7.73e5 - 2.81e5i)T^{2} \) |
| 11 | \( 1 + (-2.18e3 + 795. i)T + (1.49e7 - 1.25e7i)T^{2} \) |
| 13 | \( 1 + (-1.99e3 + 1.67e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (1.90e4 + 3.29e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.31e4 - 2.28e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.16e4 - 6.58e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-4.55e4 - 3.82e4i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (2.50e4 + 1.41e5i)T + (-2.58e10 + 9.40e9i)T^{2} \) |
| 37 | \( 1 + (-1.98e5 - 3.44e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.43e5 + 1.20e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (9.16e4 - 3.33e4i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-6.20e4 + 3.51e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 - 1.49e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (6.38e5 + 2.32e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-1.56e5 + 8.89e5i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (3.02e6 - 2.53e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-1.18e6 - 2.04e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.24e6 - 2.15e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.89e6 - 2.42e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (1.09e6 + 9.21e5i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (3.35e6 - 5.81e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.28e6 + 1.56e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−14.91650605260751269617785957233, −13.88052985941863285152990958095, −13.46225862800354328567398687426, −11.32331803286388148770010756331, −9.950302285494633369963471286808, −9.036670730512240525704006022818, −6.86238172570430500912547740881, −5.74659307894575260859455228195, −2.47017849688367178731315385313, −1.20313270766527120916303573458,
2.29020214931755272998182138980, 4.45012847801556763969745085567, 6.24015374814094250912322566316, 8.705079125581887714195467210517, 9.075832409144166552000651945884, 10.52032751883127875825221388168, 12.65058163941154111518880074585, 13.60701152389651885083635448312, 14.94939139060297052764893621800, 16.40199702346966102913938316848
