| L(s) = 1 | + (−12.0 + 4.37i)2-s + (−68.2 − 24.8i)3-s + (27.3 − 22.9i)4-s + (75.7 + 429. i)5-s + 928.·6-s + (226. + 1.28e3i)7-s + (590. − 1.02e3i)8-s + (2.36e3 + 1.98e3i)9-s + (−2.79e3 − 4.83e3i)10-s + (−3.99e3 + 6.91e3i)11-s + (−2.43e3 + 885. i)12-s + (−4.13e3 + 3.47e3i)13-s + (−8.34e3 − 1.44e4i)14-s + (5.49e3 − 3.11e4i)15-s + (−3.41e3 + 1.93e4i)16-s + (902. + 757. i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 + 0.386i)2-s + (−1.45 − 0.530i)3-s + (0.213 − 0.179i)4-s + (0.270 + 1.53i)5-s + 1.75·6-s + (0.249 + 1.41i)7-s + (0.407 − 0.706i)8-s + (1.07 + 0.905i)9-s + (−0.882 − 1.52i)10-s + (−0.904 + 1.56i)11-s + (−0.406 + 0.147i)12-s + (−0.522 + 0.438i)13-s + (−0.812 − 1.40i)14-s + (0.420 − 2.38i)15-s + (−0.208 + 1.18i)16-s + (0.0445 + 0.0373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.157675 - 0.325762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157675 - 0.325762i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (7.99e4 - 2.97e5i)T \) |
| good | 2 | \( 1 + (12.0 - 4.37i)T + (98.0 - 82.2i)T^{2} \) |
| 3 | \( 1 + (68.2 + 24.8i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-75.7 - 429. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-226. - 1.28e3i)T + (-7.73e5 + 2.81e5i)T^{2} \) |
| 11 | \( 1 + (3.99e3 - 6.91e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (4.13e3 - 3.47e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-902. - 757. i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 19 | \( 1 + (-4.84e3 - 1.76e3i)T + (6.84e8 + 5.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.39e4 - 5.88e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.16e4 + 2.01e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.70e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-4.02e5 + 3.37e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 - 3.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.73e3 - 4.72e3i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-3.45e5 + 1.96e6i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (3.24e4 - 1.83e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (7.59e5 - 6.37e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-6.60e5 - 3.74e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (8.11e4 + 2.95e4i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 - 5.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-2.60e5 - 1.47e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-3.51e6 - 2.94e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (-1.77e6 + 1.00e7i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-8.51e6 - 1.47e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−15.79542250799048987723273790522, −14.87290916094305518154726962955, −12.86793538937098073055130711256, −11.77512184627627777736167700988, −10.60040152168671959067031076989, −9.547774663116452055393000031740, −7.51553961006697907809397874456, −6.75108978090497656424731571422, −5.34434435946056799671054278567, −2.08773762353717498214773475326,
0.41343334315252287878051265405, 0.833546538869553998519865811569, 4.65222792370043029778420744038, 5.55791179409490230189922598834, 7.889186646303227266559885485773, 9.237071754057711727762078361699, 10.58643126228239735037524673645, 10.97769986286055566279159636585, 12.58848956420294018314777432392, 13.82644933908007858517259964167
