| L(s) = 1 | + (−1.67 − 1.17i)2-s + (−1.26 + 1.18i)3-s + (0.732 + 2.05i)4-s + (0.194 − 0.107i)5-s + (3.50 − 0.482i)6-s + (1.41 − 0.718i)7-s + (0.0968 − 0.348i)8-s + (0.209 − 2.99i)9-s + (−0.451 − 0.0491i)10-s + (−2.83 + 1.07i)11-s + (−3.34 − 1.73i)12-s + (−0.407 − 3.20i)13-s + (−3.21 − 0.467i)14-s + (−0.119 + 0.366i)15-s + (2.79 − 2.28i)16-s + (0.346 + 6.38i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.832i)2-s + (−0.731 + 0.682i)3-s + (0.366 + 1.02i)4-s + (0.0869 − 0.0481i)5-s + (1.43 − 0.196i)6-s + (0.535 − 0.271i)7-s + (0.0342 − 0.123i)8-s + (0.0697 − 0.997i)9-s + (−0.142 − 0.0155i)10-s + (−0.855 + 0.322i)11-s + (−0.966 − 0.499i)12-s + (−0.113 − 0.889i)13-s + (−0.858 − 0.124i)14-s + (−0.0307 + 0.0945i)15-s + (0.699 − 0.572i)16-s + (0.0840 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.559032 - 0.129493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.559032 - 0.129493i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.26 - 1.18i)T \) |
| 59 | \( 1 + (1.22 + 7.58i)T \) |
| good | 2 | \( 1 + (1.67 + 1.17i)T + (0.672 + 1.88i)T^{2} \) |
| 5 | \( 1 + (-0.194 + 0.107i)T + (2.65 - 4.23i)T^{2} \) |
| 7 | \( 1 + (-1.41 + 0.718i)T + (4.13 - 5.64i)T^{2} \) |
| 11 | \( 1 + (2.83 - 1.07i)T + (8.25 - 7.27i)T^{2} \) |
| 13 | \( 1 + (0.407 + 3.20i)T + (-12.5 + 3.25i)T^{2} \) |
| 17 | \( 1 + (-0.346 - 6.38i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (-0.0761 + 0.0721i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.934 + 2.94i)T + (-18.8 - 13.2i)T^{2} \) |
| 29 | \( 1 + (0.0731 + 0.807i)T + (-28.5 + 5.20i)T^{2} \) |
| 31 | \( 1 + (-8.98 + 2.66i)T + (25.9 - 16.9i)T^{2} \) |
| 37 | \( 1 + (-1.04 - 3.75i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (-5.57 + 6.10i)T + (-3.69 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-9.84 + 8.06i)T + (8.48 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-9.50 - 5.25i)T + (24.9 + 39.8i)T^{2} \) |
| 53 | \( 1 + (-2.96 + 0.322i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (1.75 + 1.23i)T + (20.5 + 57.4i)T^{2} \) |
| 67 | \( 1 + (-5.35 - 5.45i)T + (-1.20 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-5.57 + 3.35i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (-1.56 - 3.93i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (-9.24 - 8.14i)T + (9.95 + 78.3i)T^{2} \) |
| 83 | \( 1 + (-7.72 - 0.558i)T + (82.1 + 11.9i)T^{2} \) |
| 89 | \( 1 + (2.13 + 0.987i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-6.77 + 0.985i)T + (92.9 - 27.6i)T^{2} \) |
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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
−10.62366211939824951265499101044, −10.17915909867652181044565710663, −9.294630169066259734456147433205, −8.285564808241436195116303268871, −7.59694855450969099315730036840, −6.03120832143636570254368387911, −5.14313973444923081422500871295, −3.91848240760846391534596048216, −2.46064381862387944604232649022, −0.864232152537255222647432229733,
0.830584308639699836213444980994, 2.45229254387860013626387051840, 4.67858438878799435022083898134, 5.70532799056190625297596915430, 6.55913751914096833040958146060, 7.47309717140366817950748447214, 7.972294740553083245092237591027, 8.995246350178230610031652630264, 9.860222404047306997628644655914, 10.81582344811187000241752318373
