| L(s) = 1 | − 5·5-s − 14·7-s − 6·11-s + 68·13-s + 78·17-s − 44·19-s − 120·23-s + 25·25-s + 126·29-s + 244·31-s + 70·35-s − 304·37-s − 480·41-s − 104·43-s − 600·47-s − 147·49-s − 258·53-s + 30·55-s − 534·59-s + 362·61-s − 340·65-s + 268·67-s + 972·71-s + 470·73-s + 84·77-s − 1.24e3·79-s − 396·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.164·11-s + 1.45·13-s + 1.11·17-s − 0.531·19-s − 1.08·23-s + 1/5·25-s + 0.806·29-s + 1.41·31-s + 0.338·35-s − 1.35·37-s − 1.82·41-s − 0.368·43-s − 1.86·47-s − 3/7·49-s − 0.668·53-s + 0.0735·55-s − 1.17·59-s + 0.759·61-s − 0.648·65-s + 0.488·67-s + 1.62·71-s + 0.753·73-s + 0.124·77-s − 1.77·79-s − 0.523·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 244 T + p^{3} T^{2} \) |
| 37 | \( 1 + 304 T + p^{3} T^{2} \) |
| 41 | \( 1 + 480 T + p^{3} T^{2} \) |
| 43 | \( 1 + 104 T + p^{3} T^{2} \) |
| 47 | \( 1 + 600 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 534 T + p^{3} T^{2} \) |
| 61 | \( 1 - 362 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 972 T + p^{3} T^{2} \) |
| 73 | \( 1 - 470 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1244 T + p^{3} T^{2} \) |
| 83 | \( 1 + 396 T + p^{3} T^{2} \) |
| 89 | \( 1 + 972 T + p^{3} T^{2} \) |
| 97 | \( 1 + 46 T + p^{3} T^{2} \) |
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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
−9.792353281759657636053981582480, −8.421310134144132243852317106789, −8.168906571301881116353776002870, −6.74881896626155729834288495618, −6.20363913881512636114174349153, −5.02781413603319067801693076954, −3.78791583437946911555016635701, −3.10608834109513979569047647138, −1.44858661153723415934145873834, 0,
1.44858661153723415934145873834, 3.10608834109513979569047647138, 3.78791583437946911555016635701, 5.02781413603319067801693076954, 6.20363913881512636114174349153, 6.74881896626155729834288495618, 8.168906571301881116353776002870, 8.421310134144132243852317106789, 9.792353281759657636053981582480
