| L(s) = 1 | + (2.03 + 0.741i)2-s + (−2.69 − 1.31i)3-s + (0.538 + 0.451i)4-s + (−4.62 − 1.90i)5-s + (−4.52 − 4.67i)6-s + (−2.32 − 2.76i)7-s + (−3.57 − 6.19i)8-s + (5.56 + 7.07i)9-s + (−8.00 − 7.31i)10-s + (−8.70 + 1.53i)11-s + (−0.861 − 1.92i)12-s + (−3.15 − 8.66i)13-s + (−2.68 − 7.36i)14-s + (9.97 + 11.2i)15-s + (−3.18 − 18.0i)16-s + (9.89 − 17.1i)17-s + ⋯ |
| L(s) = 1 | + (1.01 + 0.370i)2-s + (−0.899 − 0.436i)3-s + (0.134 + 0.112i)4-s + (−0.924 − 0.381i)5-s + (−0.754 − 0.778i)6-s + (−0.332 − 0.395i)7-s + (−0.446 − 0.773i)8-s + (0.618 + 0.786i)9-s + (−0.800 − 0.731i)10-s + (−0.790 + 0.139i)11-s + (−0.0717 − 0.160i)12-s + (−0.242 − 0.666i)13-s + (−0.191 − 0.526i)14-s + (0.664 + 0.747i)15-s + (−0.198 − 1.12i)16-s + (0.581 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.364296 - 0.727699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364296 - 0.727699i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.69 + 1.31i)T \) |
| 5 | \( 1 + (4.62 + 1.90i)T \) |
| good | 2 | \( 1 + (-2.03 - 0.741i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (2.32 + 2.76i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (8.70 - 1.53i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.15 + 8.66i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-9.89 + 17.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.1 - 21.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.74 + 3.98i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (5.21 - 14.3i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (9.90 + 8.30i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (9.23 + 5.33i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (23.8 + 65.6i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-58.3 + 10.2i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (12.6 - 10.6i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 85.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (60.9 + 10.7i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-55.8 + 46.8i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-7.87 - 21.6i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-49.0 - 28.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-101. + 58.5i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (7.55 + 2.74i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (77.8 + 28.3i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-102. + 59.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-30.2 + 5.33i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.52399535410415952233235987860, −12.16880544594785280282420664247, −10.81565500288111489823263166171, −9.720186410512388071480775487767, −7.83384594111307227654621353879, −7.08168990249176935847734401138, −5.63374437864257550450950269128, −4.92032765580000074366434875024, −3.54726456740142647351784179933, −0.43810490370730035113976852606,
3.05990767898354770880351845712, 4.22840838105117091120304400223, 5.25832757346775915378297013946, 6.44893548870145377204560629277, 7.927141386932714418018807260087, 9.390880867516301212861127748655, 10.77520451283879574282970831339, 11.50460071280079577744714076102, 12.28498792463753331486756599209, 13.03274775302503330410060413460
