| L(s) = 1 | + (−1.81 + 2.16i)2-s + (0.470 + 1.29i)3-s + (−1.38 − 7.87i)4-s + (8.21 − 46.5i)5-s + (−3.65 − 1.33i)6-s + (33.5 + 58.0i)7-s + (19.5 + 11.3i)8-s + (60.5 − 50.8i)9-s + (85.9 + 102. i)10-s + (3.72 − 6.44i)11-s + (9.53 − 5.50i)12-s + (77.4 − 212. i)13-s + (−186. − 32.9i)14-s + (64.0 − 11.3i)15-s + (−60.1 + 21.8i)16-s + (12.5 + 10.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.0523 + 0.143i)3-s + (−0.0868 − 0.492i)4-s + (0.328 − 1.86i)5-s + (−0.101 − 0.0369i)6-s + (0.683 + 1.18i)7-s + (0.306 + 0.176i)8-s + (0.748 − 0.627i)9-s + (0.859 + 1.02i)10-s + (0.0307 − 0.0532i)11-s + (0.0662 − 0.0382i)12-s + (0.458 − 1.25i)13-s + (−0.952 − 0.167i)14-s + (0.284 − 0.0502i)15-s + (−0.234 + 0.0855i)16-s + (0.0434 + 0.0364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.30296 - 0.195338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30296 - 0.195338i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.81 - 2.16i)T \) |
| 19 | \( 1 + (106. - 345. i)T \) |
| good | 3 | \( 1 + (-0.470 - 1.29i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (-8.21 + 46.5i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (-33.5 - 58.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-3.72 + 6.44i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-77.4 + 212. i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-12.5 - 10.5i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (130. + 740. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-902. - 1.07e3i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (649. - 374. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.15e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-203. - 560. i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (213. - 1.20e3i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (479. - 401. i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-1.42e3 + 251. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (2.21e3 - 2.63e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-214. - 1.21e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-2.61e3 - 3.11e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-4.41e3 - 777. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (4.32e3 - 1.57e3i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (1.78e3 + 4.91e3i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (3.83e3 + 6.64e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (655. - 1.80e3i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (90.6 - 108. i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
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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
−15.73816741250423596017066901815, −14.63983431082347247766679738784, −12.88140071758248695060387370079, −12.20887218248136149340274299683, −10.14986180980849564641407583145, −8.809435220324017948361638339837, −8.284228926805326840238203254021, −5.90837453091628311524978542287, −4.78276613601693226259879113535, −1.25371590740096466627609651101,
2.00636115504829982315069902542, 3.99201205569928298793016299355, 6.81813305169054732729138511987, 7.62439665982097534449851762447, 9.752418536887908088718550750516, 10.78458076714723314242912550054, 11.39868279435623553866794244101, 13.59058725039296617131540120601, 14.05097246397931865129306932363, 15.57089912619482767761039484298
