| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s + 2·11-s − 12-s − 2·13-s + 14-s + 15-s − 16-s − 5·17-s + 18-s − 2·19-s − 20-s + 21-s + 2·22-s + 6·23-s − 3·24-s − 4·25-s − 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s + 0.426·22-s + 1.25·23-s − 0.612·24-s − 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.80242916407751, −15.36582715915772, −14.81361615824630, −14.42302423863779, −13.92263852423255, −13.39639762865953, −12.98626772071129, −12.39580677139625, −11.87108690361694, −11.06214922419092, −10.63111718891116, −9.674337951964926, −9.263885949290800, −8.899143937490584, −8.253356533843347, −7.490122758312336, −6.776464984549105, −6.237964661347901, −5.392394051421638, −4.931817305838875, −4.150715584859989, −3.805808216346850, −2.788543440178847, −2.243515390474853, −1.287784212571702, 0,
1.287784212571702, 2.243515390474853, 2.788543440178847, 3.805808216346850, 4.150715584859989, 4.931817305838875, 5.392394051421638, 6.237964661347901, 6.776464984549105, 7.490122758312336, 8.253356533843347, 8.899143937490584, 9.263885949290800, 9.674337951964926, 10.63111718891116, 11.06214922419092, 11.87108690361694, 12.39580677139625, 12.98626772071129, 13.39639762865953, 13.92263852423255, 14.42302423863779, 14.81361615824630, 15.36582715915772, 15.80242916407751
