| L(s) = 1 | + (1.69 − 2.94i)3-s + (105. − 182. i)5-s + 1.46e3·7-s + (1.08e3 + 1.88e3i)9-s − 7.03e3·11-s + (512. + 887. i)13-s + (−358. − 621. i)15-s + (1.01e4 − 1.75e4i)17-s + (2.71e4 − 1.26e4i)19-s + (2.49e3 − 4.32e3i)21-s + (1.60e4 + 2.78e4i)23-s + (1.67e4 + 2.90e4i)25-s + 1.48e4·27-s + (−6.58e4 − 1.14e5i)29-s + 2.99e5·31-s + ⋯ |
| L(s) = 1 | + (0.0363 − 0.0629i)3-s + (0.377 − 0.653i)5-s + 1.61·7-s + (0.497 + 0.861i)9-s − 1.59·11-s + (0.0646 + 0.111i)13-s + (−0.0274 − 0.0475i)15-s + (0.499 − 0.864i)17-s + (0.906 − 0.421i)19-s + (0.0588 − 0.101i)21-s + (0.275 + 0.477i)23-s + (0.214 + 0.372i)25-s + 0.144·27-s + (−0.501 − 0.868i)29-s + 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.46729 - 0.498816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46729 - 0.498816i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.71e4 + 1.26e4i)T \) |
| good | 3 | \( 1 + (-1.69 + 2.94i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-105. + 182. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 - 1.46e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-512. - 887. i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.01e4 + 1.75e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-1.60e4 - 2.78e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (6.58e4 + 1.14e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.99e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.48e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (8.31e4 - 1.43e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.65e5 + 2.85e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.91e5 + 6.78e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.89e5 - 3.27e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-6.99e5 + 1.21e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (3.73e5 + 6.46e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.45e6 + 4.24e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.76e6 - 3.05e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.66e6 + 2.89e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.74e6 - 6.47e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.14e6 - 8.90e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.88e6 + 3.25e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.29198412911574856610719342502, −11.81453224049637669455901740408, −10.84333086777425006138884228046, −9.659490227508444423513111995306, −8.141394421134527273994826133374, −7.56205833577297407731982339126, −5.26534128590879675701448709347, −4.83755848638284782735604264188, −2.41571840322833473600918202359, −1.08264685842089149501525815894,
1.24768302319554773205989807151, 2.82172788191225176807277411484, 4.60632738983183794080802361025, 5.88822645119540579327767577961, 7.45929799500009448251859793340, 8.377740527115461603303452858246, 10.04735565828419326616749338095, 10.77432555580641814199761148561, 11.98357586528953059928377760234, 13.18554108318915571262929370032
