| L(s) = 1 | + (−3.93 − 2.85i)2-s + (8.34 − 25.6i)3-s + (−32.2 − 99.2i)4-s + (−221. + 160. i)5-s + (−106. + 77.1i)6-s + (−146. − 451. i)7-s + (−348. + 1.07e3i)8-s + (−589. − 428. i)9-s + 1.33e3·10-s + (3.31e3 + 2.91e3i)11-s − 2.81e3·12-s + (592. + 430. i)13-s + (−713. + 2.19e3i)14-s + (2.28e3 + 7.02e3i)15-s + (−6.36e3 + 4.62e3i)16-s + (−1.87e4 + 1.35e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.347 − 0.252i)2-s + (0.178 − 0.549i)3-s + (−0.252 − 0.775i)4-s + (−0.792 + 0.575i)5-s + (−0.200 + 0.145i)6-s + (−0.161 − 0.497i)7-s + (−0.240 + 0.741i)8-s + (−0.269 − 0.195i)9-s + 0.420·10-s + (0.751 + 0.659i)11-s − 0.470·12-s + (0.0747 + 0.0543i)13-s + (−0.0694 + 0.213i)14-s + (0.174 + 0.537i)15-s + (−0.388 + 0.282i)16-s + (−0.923 + 0.671i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.124937 + 0.160127i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124937 + 0.160127i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.34 + 25.6i)T \) |
| 11 | \( 1 + (-3.31e3 - 2.91e3i)T \) |
| good | 2 | \( 1 + (3.93 + 2.85i)T + (39.5 + 121. i)T^{2} \) |
| 5 | \( 1 + (221. - 160. i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (146. + 451. i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-592. - 430. i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.87e4 - 1.35e4i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.33e4 - 4.12e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 - 3.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (3.77e4 + 1.16e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (2.35e5 + 1.70e5i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (9.33e4 + 2.87e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (9.19e4 - 2.83e5i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 + 2.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.83e5 - 8.72e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (5.36e5 + 3.89e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-4.70e4 - 1.44e5i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.52e6 - 1.10e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + 3.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.75e6 - 1.27e6i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (8.28e5 + 2.55e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-2.33e6 - 1.69e6i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-5.09e6 + 3.69e6i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 + 5.10e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.30e6 - 9.50e5i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−15.12783922021512829814878142077, −14.52622980852006454452597241673, −13.10233457744317157198884720203, −11.60655575082596140471041650121, −10.55753358463429466178981295872, −9.159137167036334467568765455561, −7.62266875279825538196267771591, −6.25896823214941270005947072384, −4.00509287037208707512359217375, −1.73594310067200387161956889429,
0.10361372931229933336862615744, 3.29482306005297841719088286463, 4.70641177895009639231779731982, 6.98574870266961982131879084451, 8.695872022105121142642144362633, 9.025689622103090467980593463691, 11.17328199344627280068519121470, 12.29909065268665838072918916314, 13.53805843065576313684501153639, 15.24144140514182930919756536134
