| L(s) = 1 | + (−7.79 + 13.5i)2-s + (−13.5 + 7.79i)3-s + (−89.6 − 155. i)4-s + (22.3 + 12.9i)5-s − 243. i·6-s + (−203. + 276. i)7-s + 1.79e3·8-s + (121.5 − 210. i)9-s + (−348. + 201. i)10-s + (−311. − 540. i)11-s + (2.41e3 + 1.39e3i)12-s − 3.25e3i·13-s + (−2.14e3 − 4.89e3i)14-s − 402.·15-s + (−8.27e3 + 1.43e4i)16-s + (275. − 158. i)17-s + ⋯ |
| L(s) = 1 | + (−0.974 + 1.68i)2-s + (−0.5 + 0.288i)3-s + (−1.40 − 2.42i)4-s + (0.178 + 0.103i)5-s − 1.12i·6-s + (−0.592 + 0.805i)7-s + 3.50·8-s + (0.166 − 0.288i)9-s + (−0.348 + 0.201i)10-s + (−0.234 − 0.405i)11-s + (1.40 + 0.808i)12-s − 1.48i·13-s + (−0.783 − 1.78i)14-s − 0.119·15-s + (−2.02 + 3.49i)16-s + (0.0560 − 0.0323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.126548 - 0.0505833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.126548 - 0.0505833i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.5 - 7.79i)T \) |
| 7 | \( 1 + (203. - 276. i)T \) |
| good | 2 | \( 1 + (7.79 - 13.5i)T + (-32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (-22.3 - 12.9i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (311. + 540. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 3.25e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-275. + 158. i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (5.19e3 + 3.00e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-21.0 + 36.3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.42e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (1.75e4 - 1.01e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-8.68e3 + 1.50e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 1.00e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.78e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-5.70e4 - 3.29e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (4.96e4 + 8.59e4i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-8.71e4 + 5.03e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.01e5 + 1.16e5i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.05e4 + 3.55e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.00e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-4.79e5 + 2.76e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-5.24e3 + 9.07e3i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 7.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.64e5 + 9.52e4i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.07e6iT - 8.32e11T^{2} \) |
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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
−16.59663357250783592602179475317, −15.65031816761995705455329359771, −14.84585379154219749065090676080, −13.10574733309301088339925794123, −10.62370957658713438050822607245, −9.447674631426078148180535429733, −8.102330387234781938401250412642, −6.36655293360383032363135203511, −5.36806603039245769126596831090, −0.12528835614005209793905094144,
1.76558587244550522737984977200, 4.02850297875057390650102706519, 7.32789561451154736016953824340, 9.174400056937278920682653838985, 10.31129830086727870455778204843, 11.44831060437263260574094191385, 12.64498884056846365063359182765, 13.63650055281389018728876374100, 16.62546626812571063873788134537, 17.19538794618818688979929812888
