| L(s) = 1 | + (−0.833 + 1.73i)2-s + (−5.15 − 0.385i)3-s + (7.67 + 9.62i)4-s + (0.609 − 1.97i)5-s + (4.96 − 8.59i)6-s + (−32.4 + 18.7i)7-s + (−53.0 + 12.1i)8-s + (−53.7 − 8.09i)9-s + (2.91 + 2.70i)10-s + (−114. + 144. i)11-s + (−35.8 − 52.5i)12-s + (50.0 − 46.3i)13-s + (−5.37 − 71.7i)14-s + (−3.90 + 9.94i)15-s + (−20.5 + 90.1i)16-s + (134. − 41.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.208 + 0.432i)2-s + (−0.572 − 0.0428i)3-s + (0.479 + 0.601i)4-s + (0.0243 − 0.0790i)5-s + (0.137 − 0.238i)6-s + (−0.661 + 0.382i)7-s + (−0.828 + 0.189i)8-s + (−0.663 − 0.0999i)9-s + (0.0291 + 0.0270i)10-s + (−0.949 + 1.19i)11-s + (−0.248 − 0.364i)12-s + (0.295 − 0.274i)13-s + (−0.0274 − 0.366i)14-s + (−0.0173 + 0.0441i)15-s + (−0.0803 + 0.352i)16-s + (0.463 − 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.126004 + 0.672862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.126004 + 0.672862i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (191. - 1.83e3i)T \) |
| good | 2 | \( 1 + (0.833 - 1.73i)T + (-9.97 - 12.5i)T^{2} \) |
| 3 | \( 1 + (5.15 + 0.385i)T + (80.0 + 12.0i)T^{2} \) |
| 5 | \( 1 + (-0.609 + 1.97i)T + (-516. - 352. i)T^{2} \) |
| 7 | \( 1 + (32.4 - 18.7i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (114. - 144. i)T + (-3.25e3 - 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-50.0 + 46.3i)T + (2.13e3 - 2.84e4i)T^{2} \) |
| 17 | \( 1 + (-134. + 41.3i)T + (6.90e4 - 4.70e4i)T^{2} \) |
| 19 | \( 1 + (13.7 + 91.2i)T + (-1.24e5 + 3.84e4i)T^{2} \) |
| 23 | \( 1 + (-173. - 441. i)T + (-2.05e5 + 1.90e5i)T^{2} \) |
| 29 | \( 1 + (488. - 36.5i)T + (6.99e5 - 1.05e5i)T^{2} \) |
| 31 | \( 1 + (-638. + 435. i)T + (3.37e5 - 8.59e5i)T^{2} \) |
| 37 | \( 1 + (-1.17e3 - 679. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (2.55e3 + 1.22e3i)T + (1.76e6 + 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-1.22e3 - 1.54e3i)T + (-1.08e6 + 4.75e6i)T^{2} \) |
| 53 | \( 1 + (433. + 402. i)T + (5.89e5 + 7.86e6i)T^{2} \) |
| 59 | \( 1 + (-144. + 632. i)T + (-1.09e7 - 5.25e6i)T^{2} \) |
| 61 | \( 1 + (-2.84e3 + 4.17e3i)T + (-5.05e6 - 1.28e7i)T^{2} \) |
| 67 | \( 1 + (655. - 98.7i)T + (1.92e7 - 5.93e6i)T^{2} \) |
| 71 | \( 1 + (2.55e3 + 1.00e3i)T + (1.86e7 + 1.72e7i)T^{2} \) |
| 73 | \( 1 + (-1.94e3 - 2.09e3i)T + (-2.12e6 + 2.83e7i)T^{2} \) |
| 79 | \( 1 + (-4.04e3 - 7.00e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (363. - 4.85e3i)T + (-4.69e7 - 7.07e6i)T^{2} \) |
| 89 | \( 1 + (7.23e3 + 541. i)T + (6.20e7 + 9.35e6i)T^{2} \) |
| 97 | \( 1 + (-3.95e3 + 4.95e3i)T + (-1.96e7 - 8.63e7i)T^{2} \) |
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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
−15.79728386571810005244677040573, −15.00489685803738473191395817137, −13.07849869776380934308711698137, −12.21793512048493937216042288946, −11.08819591833959116616415467625, −9.503130028159061790725525479992, −8.022340881881940813525221542151, −6.73442585995719017838533042584, −5.41031282692974712159679315718, −2.91654172428149662934696508689,
0.47667060833798299062800180214, 2.96324828785147585891342187866, 5.52709933280625610343651041166, 6.57413287880944267707444827342, 8.574262657047398245166769712447, 10.24539767235790362272050645681, 10.89995332563177334116706719668, 12.01226414789737615863396071426, 13.46448389426881266194192506510, 14.70955800012312067763758711020
