| L(s) = 1 | + 0.836·2-s + 6.41·3-s − 7.29·4-s − 1.43·5-s + 5.36·6-s − 12.8·8-s + 14.1·9-s − 1.19·10-s + 54.3·11-s − 46.8·12-s − 67.1·13-s − 9.18·15-s + 47.6·16-s − 18.9·17-s + 11.8·18-s + 58.6·19-s + 10.4·20-s + 45.4·22-s + 19.2·23-s − 82.1·24-s − 122.·25-s − 56.2·26-s − 82.3·27-s + 228.·29-s − 7.68·30-s − 165.·31-s + 142.·32-s + ⋯ |
| L(s) = 1 | + 0.295·2-s + 1.23·3-s − 0.912·4-s − 0.128·5-s + 0.365·6-s − 0.565·8-s + 0.524·9-s − 0.0378·10-s + 1.49·11-s − 1.12·12-s − 1.43·13-s − 0.158·15-s + 0.744·16-s − 0.269·17-s + 0.155·18-s + 0.707·19-s + 0.116·20-s + 0.440·22-s + 0.174·23-s − 0.698·24-s − 0.983·25-s − 0.424·26-s − 0.587·27-s + 1.46·29-s − 0.0467·30-s − 0.957·31-s + 0.786·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 0.836T + 8T^{2} \) |
| 3 | \( 1 - 6.41T + 27T^{2} \) |
| 5 | \( 1 + 1.43T + 125T^{2} \) |
| 11 | \( 1 - 54.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 228.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 246.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 531.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 167.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 619.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 396.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 981.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 413.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.06e3T + 9.12e5T^{2} \) |
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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
−8.311494356041956184155776492386, −7.66315195854786483734553506806, −6.80802986975498520701761027337, −5.79036588644457657719335543898, −4.78455031663971311526500754084, −4.10984743280913082103527373389, −3.36437435487920591967229448231, −2.55119006837694050583013425358, −1.35008612133212029048058058451, 0,
1.35008612133212029048058058451, 2.55119006837694050583013425358, 3.36437435487920591967229448231, 4.10984743280913082103527373389, 4.78455031663971311526500754084, 5.79036588644457657719335543898, 6.80802986975498520701761027337, 7.66315195854786483734553506806, 8.311494356041956184155776492386
