| L(s) = 1 | + (−5.59 − 7.01i)2-s + (18.1 − 22.7i)3-s + (10.5 − 46.3i)4-s + (−162. + 78.4i)5-s − 261.·6-s − 216.·7-s + (−1.41e3 + 683. i)8-s + (297. + 1.30e3i)9-s + (1.46e3 + 703. i)10-s + (−1.21e3 − 5.31e3i)11-s + (−863. − 1.08e3i)12-s + (−6.47e3 + 3.11e3i)13-s + (1.20e3 + 1.51e3i)14-s + (−1.17e3 + 5.13e3i)15-s + (7.23e3 + 3.48e3i)16-s + (1.17e4 + 5.65e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.494 − 0.619i)2-s + (0.388 − 0.486i)3-s + (0.0827 − 0.362i)4-s + (−0.582 + 0.280i)5-s − 0.493·6-s − 0.238·7-s + (−0.979 + 0.471i)8-s + (0.136 + 0.596i)9-s + (0.461 + 0.222i)10-s + (−0.274 − 1.20i)11-s + (−0.144 − 0.180i)12-s + (−0.816 + 0.393i)13-s + (0.117 + 0.147i)14-s + (−0.0896 + 0.392i)15-s + (0.441 + 0.212i)16-s + (0.579 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00325 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.00325 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.108076 + 0.107725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.108076 + 0.107725i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.70e4 + 5.15e5i)T \) |
| good | 2 | \( 1 + (5.59 + 7.01i)T + (-28.4 + 124. i)T^{2} \) |
| 3 | \( 1 + (-18.1 + 22.7i)T + (-486. - 2.13e3i)T^{2} \) |
| 5 | \( 1 + (162. - 78.4i)T + (4.87e4 - 6.10e4i)T^{2} \) |
| 7 | \( 1 + 216.T + 8.23e5T^{2} \) |
| 11 | \( 1 + (1.21e3 + 5.31e3i)T + (-1.75e7 + 8.45e6i)T^{2} \) |
| 13 | \( 1 + (6.47e3 - 3.11e3i)T + (3.91e7 - 4.90e7i)T^{2} \) |
| 17 | \( 1 + (-1.17e4 - 5.65e3i)T + (2.55e8 + 3.20e8i)T^{2} \) |
| 19 | \( 1 + (5.40e3 - 2.36e4i)T + (-8.05e8 - 3.87e8i)T^{2} \) |
| 23 | \( 1 + (-1.27e4 - 5.58e4i)T + (-3.06e9 + 1.47e9i)T^{2} \) |
| 29 | \( 1 + (2.45e4 + 3.07e4i)T + (-3.83e9 + 1.68e10i)T^{2} \) |
| 31 | \( 1 + (3.37e4 + 4.23e4i)T + (-6.12e9 + 2.68e10i)T^{2} \) |
| 37 | \( 1 + 3.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.03e5 + 2.55e5i)T + (-4.33e10 + 1.89e11i)T^{2} \) |
| 47 | \( 1 + (-2.27e4 + 9.95e4i)T + (-4.56e11 - 2.19e11i)T^{2} \) |
| 53 | \( 1 + (-3.82e5 - 1.84e5i)T + (7.32e11 + 9.18e11i)T^{2} \) |
| 59 | \( 1 + (1.96e6 + 9.46e5i)T + (1.55e12 + 1.94e12i)T^{2} \) |
| 61 | \( 1 + (2.50e5 - 3.14e5i)T + (-6.99e11 - 3.06e12i)T^{2} \) |
| 67 | \( 1 + (6.48e5 - 2.83e6i)T + (-5.46e12 - 2.62e12i)T^{2} \) |
| 71 | \( 1 + (-3.16e5 + 1.38e6i)T + (-8.19e12 - 3.94e12i)T^{2} \) |
| 73 | \( 1 + (4.92e6 - 2.37e6i)T + (6.88e12 - 8.63e12i)T^{2} \) |
| 79 | \( 1 + 4.70e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + (-3.59e6 + 4.50e6i)T + (-6.03e12 - 2.64e13i)T^{2} \) |
| 89 | \( 1 + (3.40e6 - 4.26e6i)T + (-9.84e12 - 4.31e13i)T^{2} \) |
| 97 | \( 1 + (2.79e6 + 1.22e7i)T + (-7.27e13 + 3.50e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.62411499343484952156303376295, −13.59152462709170877006174058203, −12.12754642368421156175123538263, −11.07735167787582271130892295402, −10.01081976681944714429732524178, −8.574955291336633481761134896558, −7.37067802074240156958487115620, −5.62169981557569962164873420385, −3.24531407175567964163116419210, −1.71812937334570133111599854999,
0.06889356026514362740353281782, 3.02405001377059851814592329471, 4.57795982516567094403678115092, 6.77240311626814207283681237240, 7.84342166094440465130577934217, 9.071954400629258275852961695660, 10.05790723304095112920981630903, 12.04441384089248802811101349357, 12.71564669873502801447138055718, 14.73070968233781076134749161878
