| L(s) = 1 | + (−1.23 + 3.80i)2-s + 13.2·3-s + (−12.9 − 9.40i)4-s + (−68.5 − 49.7i)5-s + (−16.4 + 50.5i)6-s + (50.6 + 155. i)7-s + (51.7 − 37.6i)8-s − 66.7·9-s + (274. − 199. i)10-s + (579. − 421. i)11-s + (−171. − 124. i)12-s + (198. − 611. i)13-s − 655.·14-s + (−909. − 661. i)15-s + (79.1 + 243. i)16-s + (542. − 393. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + 0.851·3-s + (−0.404 − 0.293i)4-s + (−1.22 − 0.890i)5-s + (−0.186 + 0.572i)6-s + (0.390 + 1.20i)7-s + (0.286 − 0.207i)8-s − 0.274·9-s + (0.866 − 0.629i)10-s + (1.44 − 1.04i)11-s + (−0.344 − 0.250i)12-s + (0.326 − 1.00i)13-s − 0.893·14-s + (−1.04 − 0.758i)15-s + (0.0772 + 0.237i)16-s + (0.455 − 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.44775 - 0.493414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44775 - 0.493414i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 41 | \( 1 + (8.24e3 - 6.92e3i)T \) |
| good | 3 | \( 1 - 13.2T + 243T^{2} \) |
| 5 | \( 1 + (68.5 + 49.7i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-50.6 - 155. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-579. + 421. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-198. + 611. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-542. + 393. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (470. + 1.44e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.34e3 + 4.13e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (290. + 210. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.07e3 + 2.23e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.81e3 - 2.77e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (4.69e3 - 1.44e4i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (57.8 - 178. i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-3.50e3 - 2.54e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.91e3 - 5.88e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.18e3 + 9.80e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (4.33e4 + 3.14e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.01e4 + 2.91e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + 8.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.88e4 + 5.81e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.50e4 - 2.54e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.41143732621665380245634380337, −12.14541189361531823580286209622, −11.28400915261385513368548999921, −9.130411052607626512320967208349, −8.566928624681411144448309469663, −8.015506765467303221908747122937, −6.16361287253347312803537175407, −4.71012532671754912160810018700, −3.14809684962350723928469605335, −0.68891021106699882256690526160,
1.58730600953013450333276372707, 3.58361381543712531454763277340, 4.05025764090023587401347475923, 6.98031790405947444496658941833, 7.76425128619924291035753152782, 9.000662267868726714871475408891, 10.23381774656065084121711295780, 11.39528225972367479886922326480, 11.99426388047895501172615687195, 13.72673432225426559908959211896
