| L(s) = 1 | + (0.878 + 0.0768i)2-s + (3.70 + 3.63i)3-s + (−7.11 − 1.25i)4-s + (1.02 + 11.1i)5-s + (2.97 + 3.48i)6-s + (−28.1 − 19.7i)7-s + (−12.9 − 3.47i)8-s + (0.511 + 26.9i)9-s + (0.0413 + 9.86i)10-s + (−16.3 + 44.9i)11-s + (−21.8 − 30.5i)12-s + (−6.23 − 71.2i)13-s + (−23.2 − 19.5i)14-s + (−36.7 + 45.0i)15-s + (43.1 + 15.7i)16-s + (−27.3 + 7.33i)17-s + ⋯ |
| L(s) = 1 | + (0.310 + 0.0271i)2-s + (0.713 + 0.700i)3-s + (−0.889 − 0.156i)4-s + (0.0913 + 0.995i)5-s + (0.202 + 0.236i)6-s + (−1.52 − 1.06i)7-s + (−0.573 − 0.153i)8-s + (0.0189 + 0.999i)9-s + (0.00130 + 0.311i)10-s + (−0.448 + 1.23i)11-s + (−0.524 − 0.734i)12-s + (−0.132 − 1.52i)13-s + (−0.443 − 0.372i)14-s + (−0.632 + 0.774i)15-s + (0.674 + 0.245i)16-s + (−0.390 + 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.113i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0399695 + 0.702174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0399695 + 0.702174i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.70 - 3.63i)T \) |
| 5 | \( 1 + (-1.02 - 11.1i)T \) |
| good | 2 | \( 1 + (-0.878 - 0.0768i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (28.1 + 19.7i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (16.3 - 44.9i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (6.23 + 71.2i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (27.3 - 7.33i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (19.0 - 10.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-80.9 - 115. i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (29.4 - 24.6i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (20.2 - 115. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-3.59 - 13.4i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (217. - 259. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (21.5 + 46.1i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (16.7 - 23.9i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-38.8 + 38.8i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-458. + 166. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (109. + 623. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-856. + 74.8i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (66.1 + 38.1i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (43.6 - 162. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-230. - 274. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-9.34 + 106. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (307. + 532. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (903. - 421. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−13.22359572579132201971914629449, −12.86008707679448210315507656049, −10.67686186622110467872907407276, −10.02393846070137376946836069726, −9.523819398436731335881114195645, −7.900142492491357355559389039436, −6.81428726893998431271368281084, −5.21413926595960938364721428141, −3.79510564268351976062141436274, −3.02862599466482823810241305020,
0.28667898692633485623649612816, 2.60406507950338774017789095563, 3.95290531366496784235682622112, 5.57208779252557744357546658869, 6.62711843574313219371517021121, 8.499400168102865064537056177598, 8.914443779249839922443699090164, 9.612084763062525346286174675906, 11.81142229427603849670436081062, 12.60134221418195337518965746438
