| L(s) = 1 | + (2.47 + 7.60i)2-s + (9.33 − 28.7i)3-s + (−51.7 + 37.6i)4-s − 490.·5-s + 241.·6-s + (1.15e3 − 840. i)7-s + (−414. − 300. i)8-s + (1.03e3 + 748. i)9-s + (−1.21e3 − 3.73e3i)10-s + (−794. + 577. i)11-s + (597. + 1.83e3i)12-s + (−3.00e3 + 9.24e3i)13-s + (9.25e3 + 6.72e3i)14-s + (−4.58e3 + 1.41e4i)15-s + (1.26e3 − 3.89e3i)16-s + (9.53e3 + 6.92e3i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.199 − 0.614i)3-s + (−0.404 + 0.293i)4-s − 1.75·5-s + 0.456·6-s + (1.27 − 0.926i)7-s + (−0.286 − 0.207i)8-s + (0.471 + 0.342i)9-s + (−0.383 − 1.18i)10-s + (−0.180 + 0.130i)11-s + (0.0998 + 0.307i)12-s + (−0.379 + 1.16i)13-s + (0.901 + 0.655i)14-s + (−0.350 + 1.07i)15-s + (0.0772 − 0.237i)16-s + (0.470 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0691 - 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0691 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.13478 + 1.05879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13478 + 1.05879i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (1.37e5 - 9.32e4i)T \) |
| good | 3 | \( 1 + (-9.33 + 28.7i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 5 | \( 1 + 490.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-1.15e3 + 840. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (794. - 577. i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (3.00e3 - 9.24e3i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-9.53e3 - 6.92e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.22e4 - 3.77e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (2.30e4 + 1.67e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-7.59e4 - 2.33e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 37 | \( 1 - 4.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.35e5 + 4.16e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + (9.23e4 + 2.84e5i)T + (-2.19e11 + 1.59e11i)T^{2} \) |
| 47 | \( 1 + (3.78e5 - 1.16e6i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-6.66e5 - 4.84e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-1.03e4 + 3.19e4i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 - 3.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.17e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (3.42e6 + 2.48e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.43e6 - 1.04e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (7.32e5 + 5.31e5i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-1.59e5 - 4.89e5i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (6.78e6 - 4.92e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-8.68e6 + 6.30e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−14.21940614617628768728077576106, −12.63851941721775094562853746116, −11.79322088395006322634404600567, −10.58088264195221430038593236698, −8.428140333254059634829008633633, −7.62346178599079191469027910962, −7.10742205569604205300348730867, −4.76157178405228574344128616117, −3.88819165780626820379064420469, −1.31293595311243714016208369793,
0.60636743587877344405863362190, 2.84882470059701443156159591191, 4.18160592381467061169081826938, 5.15757059961193144841722621111, 7.65030085548837575916391794332, 8.516548648143937113065175276500, 9.940573939534993474930489615590, 11.42669389877784399675852717772, 11.72319383611585378195287546768, 12.98959749900551921722699033430
