| L(s) = 1 | + (0.428 − 0.247i)2-s + (0.866 − 1.5i)3-s + (−1.87 + 3.25i)4-s + (2.23 + 4.47i)5-s − 0.856i·6-s + (5.82 + 3.87i)7-s + 3.83i·8-s + (−1.5 − 2.59i)9-s + (2.06 + 1.36i)10-s + (3.55 − 6.16i)11-s + (3.25 + 5.63i)12-s + 6.94·13-s + (3.45 + 0.219i)14-s + (8.64 + 0.516i)15-s + (−6.56 − 11.3i)16-s + (6.89 − 11.9i)17-s + ⋯ |
| L(s) = 1 | + (0.214 − 0.123i)2-s + (0.288 − 0.5i)3-s + (−0.469 + 0.813i)4-s + (0.447 + 0.894i)5-s − 0.142i·6-s + (0.832 + 0.553i)7-s + 0.479i·8-s + (−0.166 − 0.288i)9-s + (0.206 + 0.136i)10-s + (0.323 − 0.560i)11-s + (0.271 + 0.469i)12-s + 0.534·13-s + (0.246 + 0.0156i)14-s + (0.576 + 0.0344i)15-s + (−0.410 − 0.710i)16-s + (0.405 − 0.702i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.60028 + 0.431806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60028 + 0.431806i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 + (-2.23 - 4.47i)T \) |
| 7 | \( 1 + (-5.82 - 3.87i)T \) |
| good | 2 | \( 1 + (-0.428 + 0.247i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (-3.55 + 6.16i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 6.94T + 169T^{2} \) |
| 17 | \( 1 + (-6.89 + 11.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (26.4 - 15.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9.08 + 5.24i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 28.7T + 841T^{2} \) |
| 31 | \( 1 + (21.1 + 12.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (18.2 - 10.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 77.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 70.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (15.0 + 26.0i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-54.6 - 31.5i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (66.4 + 38.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.73 + 2.73i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-62.8 - 36.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 60.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (44.3 - 76.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-5.40 - 9.36i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 49.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (105. - 60.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 28.1T + 9.40e3T^{2} \) |
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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
−13.83520101613662428394959728963, −12.57292587927290665368486471540, −11.69353907086077923747777705724, −10.62719567337157889532064760717, −8.968981887703116502380353117077, −8.219406537535274773824247632926, −6.95828535391443762138986078709, −5.55419985710813088977357964262, −3.72965007871088463140351870864, −2.31086170488780485507653681265,
1.45514298545540291557478558984, 4.26183011610291151512896709049, 4.96986997484662152248800942606, 6.36670227012529850001174945559, 8.236479562892270740866168921388, 9.136491252591887270465532799886, 10.15944213475640269831879685772, 11.07838697729639574418911459095, 12.72936536186514397970750531042, 13.55162203609231757006592058585
