| L(s) = 1 | + 5-s − 2·7-s − 6·11-s − 13-s − 2·17-s − 6·19-s + 2·23-s + 25-s + 29-s − 10·31-s − 2·35-s − 2·37-s − 2·41-s − 12·43-s − 12·47-s − 3·49-s − 2·53-s − 6·55-s − 4·59-s + 2·61-s − 65-s + 10·67-s − 16·71-s − 10·73-s + 12·77-s + 14·79-s + 14·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.80·11-s − 0.277·13-s − 0.485·17-s − 1.37·19-s + 0.417·23-s + 1/5·25-s + 0.185·29-s − 1.79·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s − 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.274·53-s − 0.809·55-s − 0.520·59-s + 0.256·61-s − 0.124·65-s + 1.22·67-s − 1.89·71-s − 1.17·73-s + 1.36·77-s + 1.57·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.12303918797905, −12.66019877427817, −12.33127117547875, −11.45535906613732, −11.18315501492451, −10.54148160394933, −10.27530025178208, −9.928663845356283, −9.313380802474606, −8.808385354055530, −8.443716402634333, −7.843279131386983, −7.399121161209503, −6.852495490132033, −6.307324218278954, −6.060920946203507, −5.209310772119883, −4.975821804583155, −4.554990637485292, −3.503309422449107, −3.347458870220213, −2.654337800779480, −1.980691423197631, −1.797482870969406, −0.4807943978105975, 0,
0.4807943978105975, 1.797482870969406, 1.980691423197631, 2.654337800779480, 3.347458870220213, 3.503309422449107, 4.554990637485292, 4.975821804583155, 5.209310772119883, 6.060920946203507, 6.307324218278954, 6.852495490132033, 7.399121161209503, 7.843279131386983, 8.443716402634333, 8.808385354055530, 9.313380802474606, 9.928663845356283, 10.27530025178208, 10.54148160394933, 11.18315501492451, 11.45535906613732, 12.33127117547875, 12.66019877427817, 13.12303918797905
