| L(s) = 1 | + (−8.80 − 6.39i)3-s + (10.2 − 31.6i)5-s + (−1.54 − 2.12i)7-s + (11.5 + 35.5i)9-s + (−79.0 + 91.6i)11-s + (70.0 − 22.7i)13-s + (−293. + 213. i)15-s + (−469. − 152. i)17-s + (−259. + 357. i)19-s + 28.6i·21-s + 0.449·23-s + (−392. − 285. i)25-s + (−146. + 451. i)27-s + (432. + 595. i)29-s + (−310. − 955. i)31-s + ⋯ |
| L(s) = 1 | + (−0.977 − 0.710i)3-s + (0.411 − 1.26i)5-s + (−0.0315 − 0.0434i)7-s + (0.142 + 0.438i)9-s + (−0.653 + 0.757i)11-s + (0.414 − 0.134i)13-s + (−1.30 + 0.947i)15-s + (−1.62 − 0.528i)17-s + (−0.718 + 0.989i)19-s + 0.0648i·21-s + 0.000850·23-s + (−0.628 − 0.456i)25-s + (−0.201 + 0.619i)27-s + (0.514 + 0.708i)29-s + (−0.323 − 0.994i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0843892 + 0.444093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0843892 + 0.444093i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (79.0 - 91.6i)T \) |
| good | 3 | \( 1 + (8.80 + 6.39i)T + (25.0 + 77.0i)T^{2} \) |
| 5 | \( 1 + (-10.2 + 31.6i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (1.54 + 2.12i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-70.0 + 22.7i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (469. + 152. i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (259. - 357. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 0.449T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-432. - 595. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (310. + 955. i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-1.50e3 + 1.09e3i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (1.07e3 - 1.47e3i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 458. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (2.22e3 + 1.61e3i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (170. + 525. i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (95.0 - 69.0i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (1.23e3 + 402. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 + 404.T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-2.89e3 + 8.91e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (1.82e3 + 2.50e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (4.84e3 - 1.57e3i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (7.41e3 + 2.40e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 + 9.73e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-5.22e3 - 1.60e4i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−12.89933218150038671625879350530, −11.96853458867143968322853246884, −10.83886571279565435410116818941, −9.463132050860226867569370215134, −8.278335020880778754950781077780, −6.81376011562478601191729922229, −5.67288293443961491434231928383, −4.57496451147058717381540264933, −1.79127580635385537234275284745, −0.23586162396359487098839391569,
2.67674291143101243875243235630, 4.47308108618944993512748577512, 5.93989391974507656994998948281, 6.73841641326109616841658384257, 8.556402500216418568869901125850, 10.07390639614534766251970692188, 10.96467519905888043071627994111, 11.25276307557937655437547704958, 13.06254839130217056579806588579, 14.02022495872661873421983318673
