| L(s) = 1 | + (2.47 + 7.60i)2-s + (−10.2 + 31.5i)3-s + (−51.7 + 37.6i)4-s + 307.·5-s − 264.·6-s + (767. − 557. i)7-s + (−414. − 300. i)8-s + (881. + 640. i)9-s + (760. + 2.34e3i)10-s + (5.32e3 − 3.87e3i)11-s + (−655. − 2.01e3i)12-s + (2.12e3 − 6.53e3i)13-s + (6.14e3 + 4.46e3i)14-s + (−3.15e3 + 9.69e3i)15-s + (1.26e3 − 3.89e3i)16-s + (1.48e3 + 1.08e3i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.218 + 0.673i)3-s + (−0.404 + 0.293i)4-s + 1.10·5-s − 0.500·6-s + (0.845 − 0.614i)7-s + (−0.286 − 0.207i)8-s + (0.403 + 0.292i)9-s + (0.240 + 0.740i)10-s + (1.20 − 0.877i)11-s + (−0.109 − 0.336i)12-s + (0.268 − 0.824i)13-s + (0.598 + 0.434i)14-s + (−0.241 + 0.741i)15-s + (0.0772 − 0.237i)16-s + (0.0733 + 0.0533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.25496 + 1.60223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25496 + 1.60223i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.47 - 7.60i)T \) |
| 31 | \( 1 + (1.56e5 - 5.38e4i)T \) |
| good | 3 | \( 1 + (10.2 - 31.5i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 5 | \( 1 - 307.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-767. + 557. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (-5.32e3 + 3.87e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-2.12e3 + 6.53e3i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.48e3 - 1.08e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-4.93e3 - 1.51e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-1.43e4 - 1.04e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-1.92e3 - 5.93e3i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 37 | \( 1 - 2.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.65e4 - 1.12e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + (5.11e3 + 1.57e4i)T + (-2.19e11 + 1.59e11i)T^{2} \) |
| 47 | \( 1 + (2.77e4 - 8.52e4i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-2.76e5 - 2.00e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-8.31e5 + 2.55e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + 2.41e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.43e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (2.43e6 + 1.76e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (3.05e6 - 2.22e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (4.75e5 + 3.45e5i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (1.35e6 + 4.17e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (5.49e5 - 3.99e5i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (5.01e6 - 3.64e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.98204478940459528791372061436, −13.01713535251470842697997531208, −11.28216022216037722630590302753, −10.24566329965648037700272257697, −9.122513703296233468426984374762, −7.72703597245014879726382036128, −6.17326186306195422607014948130, −5.12741709997822878379818626305, −3.77428863743000726186008694301, −1.33194109394714545692251615243,
1.35140289089600719432910656744, 2.07961384996052508199397961863, 4.34407234558323735362045328183, 5.84000687386466109177745193547, 7.02885551591268594426246341804, 8.973414084558718868850509743789, 9.774159443385311920061866184908, 11.37165038269163418581428474613, 12.14897441442963099449395805493, 13.18302809252288083650713865683
