| L(s) = 1 | + (−2.52 + 1.83i)2-s + (4.03 + 12.4i)3-s + (−6.87 + 21.1i)4-s + (23.3 + 16.9i)5-s + (−33.0 − 23.9i)6-s + (27.3 − 84.2i)7-s + (−52.3 − 161. i)8-s + (58.5 − 42.5i)9-s − 90.3·10-s + (212. + 340. i)11-s − 290.·12-s + (92.0 − 66.8i)13-s + (85.4 + 263. i)14-s + (−116. + 359. i)15-s + (−147. − 107. i)16-s + (1.04e3 + 756. i)17-s + ⋯ |
| L(s) = 1 | + (−0.446 + 0.324i)2-s + (0.258 + 0.796i)3-s + (−0.214 + 0.661i)4-s + (0.418 + 0.304i)5-s + (−0.374 − 0.271i)6-s + (0.211 − 0.649i)7-s + (−0.289 − 0.890i)8-s + (0.241 − 0.175i)9-s − 0.285·10-s + (0.529 + 0.848i)11-s − 0.582·12-s + (0.151 − 0.109i)13-s + (0.116 + 0.358i)14-s + (−0.133 + 0.412i)15-s + (−0.144 − 0.104i)16-s + (0.873 + 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0940 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0940 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.793338 + 0.721904i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.793338 + 0.721904i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-212. - 340. i)T \) |
| good | 2 | \( 1 + (2.52 - 1.83i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-4.03 - 12.4i)T + (-196. + 142. i)T^{2} \) |
| 5 | \( 1 + (-23.3 - 16.9i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-27.3 + 84.2i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-92.0 + 66.8i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.04e3 - 756. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (750. + 2.31e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + 1.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (485. - 1.49e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (7.25e3 - 5.27e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.91e3 + 1.20e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (5.31e3 + 1.63e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 6.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.43e3 - 1.36e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-4.54e3 + 3.30e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (3.56e3 - 1.09e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.49e3 - 1.08e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 3.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.55e4 + 2.58e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.74e3 - 1.45e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.56e4 - 5.49e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (2.33e4 + 1.69e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 3.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-9.79e4 + 7.11e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−20.00103447613079284486038160315, −18.09626477479462344091463804773, −17.15332667442369577791688772020, −15.81251656417348905028800885604, −14.38912502084316508279484358381, −12.61849309963420291658513056795, −10.35883682455378318243874895091, −9.066862145636643198140175089249, −7.13712918797506375393721129213, −4.01423545675127135897799127501,
1.60095846646000481142378934824, 5.81093701598285204352749281147, 8.294071894981530447325193311242, 9.825420829376359655127073946436, 11.72526467742958516071606968502, 13.44374234969829968551248770128, 14.65398222727883166190207803910, 16.70435837766785436349930791087, 18.51649324057360317803799078487, 18.78948358324460977502595731689
