| L(s) = 1 | + 2·2-s + 1.27·3-s + 4·4-s − 19.8·5-s + 2.54·6-s − 7·7-s + 8·8-s − 25.3·9-s − 39.6·10-s − 11·11-s + 5.09·12-s − 35.4·13-s − 14·14-s − 25.2·15-s + 16·16-s + 65.1·17-s − 50.7·18-s − 83.0·19-s − 79.2·20-s − 8.92·21-s − 22·22-s − 70.7·23-s + 10.1·24-s + 268.·25-s − 70.9·26-s − 66.7·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.245·3-s + 0.5·4-s − 1.77·5-s + 0.173·6-s − 0.377·7-s + 0.353·8-s − 0.939·9-s − 1.25·10-s − 0.301·11-s + 0.122·12-s − 0.756·13-s − 0.267·14-s − 0.435·15-s + 0.250·16-s + 0.929·17-s − 0.664·18-s − 1.00·19-s − 0.886·20-s − 0.0927·21-s − 0.213·22-s − 0.641·23-s + 0.0867·24-s + 2.14·25-s − 0.534·26-s − 0.475·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 1.27T + 27T^{2} \) |
| 5 | \( 1 + 19.8T + 125T^{2} \) |
| 13 | \( 1 + 35.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 243.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 482.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 573.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 884.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 414.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 205.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 584.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 629.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 320.T + 9.12e5T^{2} \) |
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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
−12.08186857954581342369888986437, −11.33785778255541022429934702104, −10.19415142221887800183285493241, −8.502108903047481483106406417401, −7.80830007971103290511042267512, −6.63751162945003884801000888918, −5.09912666178050360878131460046, −3.86343096282537823062934228500, −2.85803096797444541398220067516, 0,
2.85803096797444541398220067516, 3.86343096282537823062934228500, 5.09912666178050360878131460046, 6.63751162945003884801000888918, 7.80830007971103290511042267512, 8.502108903047481483106406417401, 10.19415142221887800183285493241, 11.33785778255541022429934702104, 12.08186857954581342369888986437
