| L(s) = 1 | + (0.0560 − 0.209i)2-s + (1.34 − 5.01i)3-s + (6.88 + 3.97i)4-s + (−10.0 − 10.0i)5-s + (−0.974 − 0.562i)6-s + (17.5 − 4.68i)7-s + (2.44 − 2.44i)8-s + (−23.3 − 13.4i)9-s + (−2.66 + 1.53i)10-s + (39.2 + 10.5i)11-s + (29.2 − 29.2i)12-s + (−30.4 + 35.6i)13-s − 3.92i·14-s + (−63.9 + 36.9i)15-s + (31.4 + 54.4i)16-s + (−58.9 + 102. i)17-s + ⋯ |
| L(s) = 1 | + (0.0198 − 0.0739i)2-s + (0.258 − 0.965i)3-s + (0.860 + 0.497i)4-s + (−0.898 − 0.898i)5-s + (−0.0662 − 0.0382i)6-s + (0.944 − 0.253i)7-s + (0.107 − 0.107i)8-s + (−0.866 − 0.499i)9-s + (−0.0841 + 0.0486i)10-s + (1.07 + 0.288i)11-s + (0.702 − 0.703i)12-s + (−0.649 + 0.760i)13-s − 0.0748i·14-s + (−1.10 + 0.635i)15-s + (0.491 + 0.850i)16-s + (−0.840 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.33073 - 0.721899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33073 - 0.721899i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.34 + 5.01i)T \) |
| 13 | \( 1 + (30.4 - 35.6i)T \) |
| good | 2 | \( 1 + (-0.0560 + 0.209i)T + (-6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (10.0 + 10.0i)T + 125iT^{2} \) |
| 7 | \( 1 + (-17.5 + 4.68i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-39.2 - 10.5i)T + (1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (58.9 - 102. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8.33 + 31.0i)T + (-5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-37.9 - 65.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-106. + 61.4i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-49.4 + 49.4i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-33.4 + 124. i)T + (-4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-77.5 + 289. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (311. + 180. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (324. - 324. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 207. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (43.3 + 161. i)T + (-1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-52.8 + 91.4i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-618. - 165. i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-234. + 62.9i)T + (3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (305. + 305. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 452.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (646. + 646. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (986. + 264. i)T + (6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-269. - 1.00e3i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−15.50820281252039408033991393152, −14.41345831406126139187752771039, −12.85979096490972640171690183233, −11.96016397682467310422224941838, −11.29096721228537647973368903542, −8.761684060625309011674616506291, −7.81873491329457044758878981864, −6.67511042487693666211177897712, −4.18288097928323793516407496190, −1.70722795812059052653972136371,
2.93849283692716647342465086036, 4.89258373976672956445530492443, 6.81455735116807735524408936868, 8.259579285359519348596056548124, 9.977474340632615053594085150402, 11.25252921868877163511127957508, 11.62151470968259361919495027690, 14.31489163638377989265190597945, 14.84159532678531389705739451235, 15.63180791027488444092216528647
