| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s + 4·19-s − 8·23-s − 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s + 2·45-s − 7·49-s − 2·53-s + 8·55-s − 4·57-s − 4·59-s + 2·61-s − 4·65-s + 4·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s − 0.274·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.49087773919892, −15.99633748301758, −15.41845636346975, −14.60044054245105, −14.18371932720275, −13.70668233877805, −13.14951520798310, −12.29966189046609, −11.95890114392885, −11.48056561580270, −10.73781930283072, −9.984088692506675, −9.646830925040174, −9.248365896146778, −8.259387310160539, −7.703185711167563, −6.819318573578103, −6.432747659713858, −5.710596573061130, −5.300281795275805, −4.360132705504210, −3.813057485513455, −2.811774165493996, −1.864017865141919, −1.298455202621086, 0,
1.298455202621086, 1.864017865141919, 2.811774165493996, 3.813057485513455, 4.360132705504210, 5.300281795275805, 5.710596573061130, 6.432747659713858, 6.819318573578103, 7.703185711167563, 8.259387310160539, 9.248365896146778, 9.646830925040174, 9.984088692506675, 10.73781930283072, 11.48056561580270, 11.95890114392885, 12.29966189046609, 13.14951520798310, 13.70668233877805, 14.18371932720275, 14.60044054245105, 15.41845636346975, 15.99633748301758, 16.49087773919892
