| L(s) = 1 | + (2.75 − 8.47i)2-s + (−21.8 + 15.8i)3-s + (39.3 + 28.5i)4-s + (−42.2 − 130. i)5-s + (74.3 + 228. i)6-s + (29.1 + 21.1i)7-s + (1.27e3 − 924. i)8-s + (225. − 693. i)9-s − 1.21e3·10-s + (4.38e3 − 512. i)11-s − 1.31e3·12-s + (4.14e3 − 1.27e4i)13-s + (259. − 188. i)14-s + (2.98e3 + 2.16e3i)15-s + (−2.41e3 − 7.42e3i)16-s + (−9.61e3 − 2.96e4i)17-s + ⋯ |
| L(s) = 1 | + (0.243 − 0.749i)2-s + (−0.467 + 0.339i)3-s + (0.307 + 0.223i)4-s + (−0.151 − 0.465i)5-s + (0.140 + 0.432i)6-s + (0.0321 + 0.0233i)7-s + (0.879 − 0.638i)8-s + (0.103 − 0.317i)9-s − 0.385·10-s + (0.993 − 0.116i)11-s − 0.219·12-s + (0.523 − 1.61i)13-s + (0.0253 − 0.0183i)14-s + (0.228 + 0.165i)15-s + (−0.147 − 0.453i)16-s + (−0.474 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.48621 - 1.26279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48621 - 1.26279i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (21.8 - 15.8i)T \) |
| 11 | \( 1 + (-4.38e3 + 512. i)T \) |
| good | 2 | \( 1 + (-2.75 + 8.47i)T + (-103. - 75.2i)T^{2} \) |
| 5 | \( 1 + (42.2 + 130. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-29.1 - 21.1i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 13 | \( 1 + (-4.14e3 + 1.27e4i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (9.61e3 + 2.96e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (3.89e4 - 2.82e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 - 7.65e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.15e5 - 8.40e4i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 31 | \( 1 + (7.01e3 - 2.15e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.12e5 - 1.54e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (4.78e5 - 3.47e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + 7.50e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-8.57e4 + 6.22e4i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (2.81e5 - 8.66e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.04e6 - 7.60e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (3.76e5 + 1.15e6i)T + (-2.54e12 + 1.84e12i)T^{2} \) |
| 67 | \( 1 + 1.79e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-3.42e5 - 1.05e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.87e5 - 1.36e5i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.18e6 - 3.63e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (7.55e5 + 2.32e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 - 1.91e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.45e6 - 1.37e7i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−15.01733099416388489384637649986, −13.24979528625317398950975881394, −12.28616656819327276246286126493, −11.26056605443127722431932568973, −10.20973570643784993219989181731, −8.498000323095459770031913460806, −6.66517706236851689533285514918, −4.76131040623494084778652780220, −3.19579955984259281438958562572, −0.991795955227047020075981783016,
1.72805446290202602884547002469, 4.44914989749901818195996000405, 6.41674520818791755580876090108, 6.83671769452395325002404860205, 8.725869646507834646086587776566, 10.76453762337768585821791659603, 11.54646208508990859381057485718, 13.22340715585001090544060674639, 14.51203428186872657698767660119, 15.30543646962170914732569557067
