| L(s) = 1 | + (−2.14 − 1.55i)2-s + (−8.34 + 25.6i)3-s + (−37.3 − 115. i)4-s + (121. − 88.0i)5-s + (57.8 − 42.0i)6-s + (231. + 712. i)7-s + (−203. + 627. i)8-s + (−589. − 428. i)9-s − 396.·10-s + (−4.13e3 + 1.53e3i)11-s + 3.26e3·12-s + (−1.23e4 − 9.00e3i)13-s + (612. − 1.88e3i)14-s + (1.25e3 + 3.84e3i)15-s + (−1.11e4 + 8.07e3i)16-s + (−6.58e3 + 4.78e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.189 − 0.137i)2-s + (−0.178 + 0.549i)3-s + (−0.292 − 0.898i)4-s + (0.433 − 0.315i)5-s + (0.109 − 0.0794i)6-s + (0.254 + 0.784i)7-s + (−0.140 + 0.432i)8-s + (−0.269 − 0.195i)9-s − 0.125·10-s + (−0.937 + 0.347i)11-s + 0.545·12-s + (−1.56 − 1.13i)13-s + (0.0596 − 0.183i)14-s + (0.0956 + 0.294i)15-s + (−0.678 + 0.492i)16-s + (−0.325 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00706512 + 0.0562971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00706512 + 0.0562971i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.34 - 25.6i)T \) |
| 11 | \( 1 + (4.13e3 - 1.53e3i)T \) |
| good | 2 | \( 1 + (2.14 + 1.55i)T + (39.5 + 121. i)T^{2} \) |
| 5 | \( 1 + (-121. + 88.0i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-231. - 712. i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (1.23e4 + 9.00e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (6.58e3 - 4.78e3i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.00e4 - 3.08e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + 1.95e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (7.96e4 + 2.45e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-1.18e5 - 8.63e4i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-9.91e4 - 3.05e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (8.89e3 - 2.73e4i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 + 4.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.51e5 + 1.08e6i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.26e6 + 9.17e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-5.19e5 - 1.60e6i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-1.70e5 + 1.23e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + 2.84e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.79e6 + 1.30e6i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-5.29e5 - 1.62e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-3.85e4 - 2.79e4i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-8.13e5 + 5.91e5i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 - 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.60e6 + 1.89e6i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−15.32621795394076485561596269031, −14.93279475490606394816846549149, −13.31796249731453510789513724555, −11.93776908954040755222271271085, −10.30331497891368994763482781146, −9.722885114364348383482954740329, −8.184652294141843171608153553205, −5.75856727472891537671746321942, −4.93265063249080470199996548998, −2.21749296380536583318617303254,
0.02655789563587924760044146990, 2.51876762699708307060581693053, 4.65117121713325643144219274488, 6.83553634991259170183747741266, 7.74151674506931631864476592306, 9.320652907130529844077424014503, 10.90304582095333333141828087499, 12.28588332868307959526001413146, 13.39788211319408435924350585593, 14.31326637059390615665937541995
