| L(s) = 1 | + (−0.132 + 0.408i)2-s + (0.0617 + 0.0448i)3-s + (6.32 + 4.59i)4-s + (11.1 − 0.672i)5-s + (−0.0265 + 0.0192i)6-s − 16.1·7-s + (−5.49 + 3.99i)8-s + (−8.34 − 25.6i)9-s + (−1.20 + 4.64i)10-s + (−10.1 + 31.2i)11-s + (0.184 + 0.567i)12-s + (−24.0 − 74.0i)13-s + (2.14 − 6.60i)14-s + (0.719 + 0.459i)15-s + (18.4 + 56.6i)16-s + (−57.3 + 41.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.0469 + 0.144i)2-s + (0.0118 + 0.00863i)3-s + (0.790 + 0.574i)4-s + (0.998 − 0.0601i)5-s + (−0.00180 + 0.00131i)6-s − 0.872·7-s + (−0.242 + 0.176i)8-s + (−0.308 − 0.950i)9-s + (−0.0381 + 0.147i)10-s + (−0.278 + 0.856i)11-s + (0.00443 + 0.0136i)12-s + (−0.513 − 1.58i)13-s + (0.0409 − 0.126i)14-s + (0.0123 + 0.00790i)15-s + (0.287 + 0.885i)16-s + (−0.818 + 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.27948 + 0.240509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27948 + 0.240509i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-11.1 + 0.672i)T \) |
| good | 2 | \( 1 + (0.132 - 0.408i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (-0.0617 - 0.0448i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 16.1T + 343T^{2} \) |
| 11 | \( 1 + (10.1 - 31.2i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (24.0 + 74.0i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (57.3 - 41.6i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-80.7 + 58.6i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (14.5 - 44.7i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-16.9 - 12.2i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (120. - 87.8i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (17.1 + 52.7i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-10.6 - 32.8i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-483. - 351. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-392. - 284. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-91.3 - 281. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-153. + 473. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (419. - 304. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (701. + 509. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-50.3 + 155. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (342. + 248. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-331. + 240. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-353. + 1.08e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.12e3 - 819. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.45799353556006171659201813867, −15.88388910459076527492605182861, −14.96472705796157482943461811775, −13.11159008171128500255348112076, −12.31198001753238900208952521746, −10.48829953676799756378651082243, −9.187958080795247944332698422076, −7.23369753059134527462879569225, −5.88330497880158144281340211131, −2.89336262368824282010079096328,
2.37397071355858829273566829855, 5.60877460017540028240848045781, 6.91798548109444081743344590002, 9.259447231485986337989410105636, 10.41373377219410829152237866090, 11.66659975129374669223594143051, 13.47287336689983163298868536616, 14.31748418203117369947990096074, 16.13372170036244843506158894587, 16.66310790521876423193816718307
