| 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
−15.30543646962170914732569557067, −14.51203428186872657698767660119, −13.22340715585001090544060674639, −11.54646208508990859381057485718, −10.76453762337768585821791659603, −8.725869646507834646086587776566, −6.83671769452395325002404860205, −6.41674520818791755580876090108, −4.44914989749901818195996000405, −1.72805446290202602884547002469,
0.991795955227047020075981783016, 3.19579955984259281438958562572, 4.76131040623494084778652780220, 6.66517706236851689533285514918, 8.498000323095459770031913460806, 10.20973570643784993219989181731, 11.26056605443127722431932568973, 12.28616656819327276246286126493, 13.24979528625317398950975881394, 15.01733099416388489384637649986
