| L(s) = 1 | − 2-s − 1.13·3-s + 4-s − 5-s + 1.13·6-s − 1.36·7-s − 8-s − 1.71·9-s + 10-s − 4.03·11-s − 1.13·12-s + 13-s + 1.36·14-s + 1.13·15-s + 16-s − 0.934·17-s + 1.71·18-s + 4.43·19-s − 20-s + 1.54·21-s + 4.03·22-s + 0.0668·23-s + 1.13·24-s + 25-s − 26-s + 5.34·27-s − 1.36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.654·3-s + 0.5·4-s − 0.447·5-s + 0.462·6-s − 0.514·7-s − 0.353·8-s − 0.572·9-s + 0.316·10-s − 1.21·11-s − 0.327·12-s + 0.277·13-s + 0.363·14-s + 0.292·15-s + 0.250·16-s − 0.226·17-s + 0.404·18-s + 1.01·19-s − 0.223·20-s + 0.336·21-s + 0.860·22-s + 0.0139·23-s + 0.231·24-s + 0.200·25-s − 0.196·26-s + 1.02·27-s − 0.257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 1.13T + 3T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 17 | \( 1 + 0.934T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 - 0.0668T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 + 7.18T + 67T^{2} \) |
| 71 | \( 1 + 5.63T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 - 6.85T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 + 0.160T + 97T^{2} \) |
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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
−7.926441707077330819320643460645, −7.58076555098309087163365037876, −6.59514125691738917453748260105, −5.93593417707938584664918298361, −5.28049688302431370416690303365, −4.34205434424767594398605082917, −3.10856863040098657331200267781, −2.58712737449721497903583819479, −0.992673365017920534404414142596, 0,
0.992673365017920534404414142596, 2.58712737449721497903583819479, 3.10856863040098657331200267781, 4.34205434424767594398605082917, 5.28049688302431370416690303365, 5.93593417707938584664918298361, 6.59514125691738917453748260105, 7.58076555098309087163365037876, 7.926441707077330819320643460645
