| L(s) = 1 | + (6.71 + 20.6i)2-s + (21.8 + 15.8i)3-s + (−278. + 202. i)4-s + (−94.4 + 290. i)5-s + (−181. + 557. i)6-s + (945. − 686. i)7-s + (−3.79e3 − 2.75e3i)8-s + (225. + 693. i)9-s − 6.63e3·10-s + (3.02e3 + 3.21e3i)11-s − 9.28e3·12-s + (−1.26e3 − 3.90e3i)13-s + (2.05e4 + 1.49e4i)14-s + (−6.67e3 + 4.84e3i)15-s + (1.78e4 − 5.50e4i)16-s + (−4.20e3 + 1.29e4i)17-s + ⋯ |
| L(s) = 1 | + (0.593 + 1.82i)2-s + (0.467 + 0.339i)3-s + (−2.17 + 1.57i)4-s + (−0.337 + 1.03i)5-s + (−0.342 + 1.05i)6-s + (1.04 − 0.756i)7-s + (−2.62 − 1.90i)8-s + (0.103 + 0.317i)9-s − 2.09·10-s + (0.685 + 0.728i)11-s − 1.55·12-s + (−0.160 − 0.492i)13-s + (1.99 + 1.45i)14-s + (−0.510 + 0.370i)15-s + (1.09 − 3.36i)16-s + (−0.207 + 0.638i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.776859 - 2.09797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.776859 - 2.09797i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-21.8 - 15.8i)T \) |
| 11 | \( 1 + (-3.02e3 - 3.21e3i)T \) |
| good | 2 | \( 1 + (-6.71 - 20.6i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (94.4 - 290. i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-945. + 686. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (1.26e3 + 3.90e3i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (4.20e3 - 1.29e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (1.93e4 + 1.40e4i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 - 8.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.82e5 - 1.32e5i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-1.22e4 - 3.78e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (1.98e5 - 1.44e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (2.54e5 + 1.85e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 5.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-8.24e5 - 5.98e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (1.21e5 + 3.73e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-8.04e5 + 5.84e5i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-7.71e4 + 2.37e5i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 - 4.41e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.15e6 + 3.54e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (7.53e5 - 5.47e5i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (5.86e5 + 1.80e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-1.38e6 + 4.25e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 - 8.29e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.67e6 - 5.14e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.38094250170112495428155040832, −14.81695314970389966768243772793, −14.21214724503340103405662285359, −12.87030967662758599945662725031, −10.83298534689375388629698172037, −8.928442191498574039481218245732, −7.59363058533174813513246110199, −6.85120662642679192817349737150, −4.89607036890421190929056977861, −3.68122778426869123217210520305,
0.901906946510647576193750321899, 2.19510651115223520811652916868, 4.03655276725606906376212689334, 5.30712065728391192072983096026, 8.569959503800343457318298447504, 9.231024265583642325276838819712, 11.19574544115447732744061450174, 11.93261314022900465505719628513, 12.90467333514490345962060810435, 14.01545290748368591050958852856
