| L(s) = 1 | + (−4.18 + 7.24i)2-s + (−46.5 − 4.69i)3-s + (28.9 + 50.1i)4-s + (47.2 + 81.8i)5-s + (228. − 317. i)6-s + (−171.5 + 297. i)7-s − 1.55e3·8-s + (2.14e3 + 436. i)9-s − 790.·10-s + (−3.76e3 + 6.51e3i)11-s + (−1.11e3 − 2.47e3i)12-s + (5.72e3 + 9.91e3i)13-s + (−1.43e3 − 2.48e3i)14-s + (−1.81e3 − 4.02e3i)15-s + (2.80e3 − 4.85e3i)16-s + 1.59e4·17-s + ⋯ |
| L(s) = 1 | + (−0.369 + 0.640i)2-s + (−0.994 − 0.100i)3-s + (0.226 + 0.392i)4-s + (0.169 + 0.292i)5-s + (0.432 − 0.600i)6-s + (−0.188 + 0.327i)7-s − 1.07·8-s + (0.979 + 0.199i)9-s − 0.250·10-s + (−0.852 + 1.47i)11-s + (−0.185 − 0.412i)12-s + (0.722 + 1.25i)13-s + (−0.139 − 0.242i)14-s + (−0.138 − 0.308i)15-s + (0.171 − 0.296i)16-s + 0.789·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.222800 - 0.327336i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.222800 - 0.327336i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (46.5 + 4.69i)T \) |
| 7 | \( 1 + (171.5 - 297. i)T \) |
| good | 2 | \( 1 + (4.18 - 7.24i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-47.2 - 81.8i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (3.76e3 - 6.51e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.72e3 - 9.91e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.59e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (4.27e4 + 7.40e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.03e3 - 3.52e3i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-9.61e3 - 1.66e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.78e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.60e4 + 9.71e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.27e5 + 2.21e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (2.86e5 - 4.95e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.44e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.37e6 + 2.38e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (6.41e5 - 1.11e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.97e6 - 3.42e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-3.36e6 + 5.82e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.24e6 - 5.61e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 8.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-3.21e6 + 5.56e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−14.55091142305569291353607473123, −12.74111000157605755337996769729, −12.19147961494053552683045728436, −10.86543244104647200079751716697, −9.693635522715906406100492686624, −8.150598209545749346900239316519, −6.87023684262995714323726567818, −6.15814047794797518466247762740, −4.44341645406027649200038875025, −2.19568258221450216734634513762,
0.19887850184247835096704003094, 1.21831551688700901783491181461, 3.30716397675569232821474390369, 5.46368356546174798862983366294, 6.13854263180255882285850425161, 8.062487174228664314839230080507, 9.655559712735681040868375261445, 10.70456811206095860713385733225, 11.15800937258386669040849953974, 12.54893052762025233244416397631
