| L(s) = 1 | + (−7.83 + 5.69i)2-s + (−10.5 + 32.6i)4-s + (−9.68 − 7.03i)5-s + (520. − 1.60e3i)7-s + (−485. − 1.49e3i)8-s + 115.·10-s + (−1.18e3 + 4.25e3i)11-s + (−1.16e3 + 846. i)13-s + (5.04e3 + 1.55e4i)14-s + (8.75e3 + 6.35e3i)16-s + (4.28e3 + 3.11e3i)17-s + (5.05e3 + 1.55e4i)19-s + (332. − 241. i)20-s + (−1.49e4 − 4.00e4i)22-s + 1.52e4·23-s + ⋯ |
| L(s) = 1 | + (−0.692 + 0.502i)2-s + (−0.0827 + 0.254i)4-s + (−0.0346 − 0.0251i)5-s + (0.573 − 1.76i)7-s + (−0.335 − 1.03i)8-s + 0.0366·10-s + (−0.267 + 0.963i)11-s + (−0.147 + 0.106i)13-s + (0.490 + 1.51i)14-s + (0.534 + 0.388i)16-s + (0.211 + 0.153i)17-s + (0.168 + 0.519i)19-s + (0.00928 − 0.00674i)20-s + (−0.299 − 0.801i)22-s + 0.262·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0195i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00254892 - 0.260560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00254892 - 0.260560i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.18e3 - 4.25e3i)T \) |
| good | 2 | \( 1 + (7.83 - 5.69i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (9.68 + 7.03i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-520. + 1.60e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (1.16e3 - 846. i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-4.28e3 - 3.11e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-5.05e3 - 1.55e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (3.42e4 - 1.05e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.03e5 - 7.49e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (1.47e5 - 4.53e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (3.03e4 + 9.35e4i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 3.56e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.37e5 + 1.04e6i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (7.93e5 - 5.76e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (5.53e5 - 1.70e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.08e5 - 1.51e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.75e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (6.48e5 + 4.71e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-8.75e5 + 2.69e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (5.16e6 - 3.75e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-2.14e6 - 1.56e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 7.10e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-3.30e6 + 2.39e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.09182698796660209286759289208, −12.00847774479021644564142702562, −10.52186533897261870059232039389, −9.854741242130404431558307663901, −8.392743604153409910859266945978, −7.49481578107569218145362621438, −6.80873567037235991219787238561, −4.73417207482149478453218457859, −3.62321641245551579541331046804, −1.35376435062489449572551278447,
0.10680731885159867820978324712, 1.74405376620990236352865435940, 2.88180245275469448707169314315, 5.18027694715959924574557835853, 5.89109330670213454651115588678, 7.943474188055012989452349974560, 8.882026452139020107030734918252, 9.585160057427800765579197323490, 11.11677328269943853020606795873, 11.52487743459088201211987581995
