| L(s) = 1 | + (1.96 + 1.56i)2-s + (0.129 − 0.102i)3-s + (0.512 + 2.24i)4-s + (−2.07 + 4.30i)5-s + 0.414·6-s − 12.3i·7-s + (1.84 − 3.83i)8-s + (−1.99 + 8.74i)9-s + (−10.8 + 5.20i)10-s + (−0.816 + 3.57i)11-s + (0.297 + 0.237i)12-s + (−10.2 − 4.93i)13-s + (19.3 − 24.2i)14-s + (0.175 + 0.769i)15-s + (17.9 − 8.63i)16-s + (−7.47 + 3.59i)17-s + ⋯ |
| L(s) = 1 | + (0.981 + 0.782i)2-s + (0.0430 − 0.0343i)3-s + (0.128 + 0.561i)4-s + (−0.414 + 0.861i)5-s + 0.0691·6-s − 1.76i·7-s + (0.230 − 0.479i)8-s + (−0.221 + 0.971i)9-s + (−1.08 + 0.520i)10-s + (−0.0742 + 0.325i)11-s + (0.0247 + 0.0197i)12-s + (−0.788 − 0.379i)13-s + (1.38 − 1.73i)14-s + (0.0117 + 0.0513i)15-s + (1.12 − 0.539i)16-s + (−0.439 + 0.211i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44991 + 0.631899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44991 + 0.631899i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-39.5 - 16.9i)T \) |
| good | 2 | \( 1 + (-1.96 - 1.56i)T + (0.890 + 3.89i)T^{2} \) |
| 3 | \( 1 + (-0.129 + 0.102i)T + (2.00 - 8.77i)T^{2} \) |
| 5 | \( 1 + (2.07 - 4.30i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + 12.3iT - 49T^{2} \) |
| 11 | \( 1 + (0.816 - 3.57i)T + (-109. - 52.4i)T^{2} \) |
| 13 | \( 1 + (10.2 + 4.93i)T + (105. + 132. i)T^{2} \) |
| 17 | \( 1 + (7.47 - 3.59i)T + (180. - 225. i)T^{2} \) |
| 19 | \( 1 + (-14.1 + 3.22i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (5.64 - 24.7i)T + (-476. - 229. i)T^{2} \) |
| 29 | \( 1 + (-10.6 - 8.47i)T + (187. + 819. i)T^{2} \) |
| 31 | \( 1 + (30.6 - 38.4i)T + (-213. - 936. i)T^{2} \) |
| 37 | \( 1 + 51.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-16.2 + 20.3i)T + (-374. - 1.63e3i)T^{2} \) |
| 47 | \( 1 + (17.7 + 77.7i)T + (-1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-20.7 + 9.98i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 + (-31.5 + 15.2i)T + (2.17e3 - 2.72e3i)T^{2} \) |
| 61 | \( 1 + (-3.30 + 2.63i)T + (828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (13.8 + 60.5i)T + (-4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-53.7 + 12.2i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (56.3 - 117. i)T + (-3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 - 48.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + (42.6 + 53.5i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (92.7 - 73.9i)T + (1.76e3 - 7.72e3i)T^{2} \) |
| 97 | \( 1 + (-22.2 + 97.4i)T + (-8.47e3 - 4.08e3i)T^{2} \) |
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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
−15.72014917763405977376912308576, −14.47939722323546287861650123825, −13.91555771408463499464157687485, −12.88643476595868187438025548276, −11.01997841611316701314199903583, −10.14683722010161386259097685868, −7.45728932081081389563606257600, −7.14303122947347376684417579109, −5.14358337438266109992513688258, −3.69952209343746022923771592275,
2.70256882886682779946949823365, 4.54110297727002654518539535869, 5.85096150037384773202134418954, 8.344753166443168791429866650905, 9.380142148921994640431821713370, 11.52908725629906538641262935979, 12.13541736310944023528601763405, 12.80925274172941910768786568471, 14.36603443824234163802676825670, 15.29374019807310069542003166059
