| L(s) = 1 | + 3.47·3-s − 20.8·5-s − 23.5·7-s − 14.9·9-s + 11·11-s + 27.1·13-s − 72.5·15-s + 63.0·17-s − 53.7·19-s − 81.6·21-s − 162.·23-s + 311.·25-s − 145.·27-s − 233.·29-s + 134.·31-s + 38.1·33-s + 491.·35-s + 303.·37-s + 94.4·39-s + 67.8·41-s − 311.·43-s + 312.·45-s − 19.5·47-s + 210.·49-s + 219.·51-s − 467.·53-s − 229.·55-s + ⋯ |
| L(s) = 1 | + 0.668·3-s − 1.86·5-s − 1.27·7-s − 0.553·9-s + 0.301·11-s + 0.580·13-s − 1.24·15-s + 0.899·17-s − 0.649·19-s − 0.848·21-s − 1.47·23-s + 2.49·25-s − 1.03·27-s − 1.49·29-s + 0.778·31-s + 0.201·33-s + 2.37·35-s + 1.34·37-s + 0.387·39-s + 0.258·41-s − 1.10·43-s + 1.03·45-s − 0.0605·47-s + 0.613·49-s + 0.601·51-s − 1.21·53-s − 0.563·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 3.47T + 27T^{2} \) |
| 5 | \( 1 + 20.8T + 125T^{2} \) |
| 7 | \( 1 + 23.5T + 343T^{2} \) |
| 13 | \( 1 - 27.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 67.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 19.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 467.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 601.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 193.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 431.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 686.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 367.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 740.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 970.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 618.T + 9.12e5T^{2} \) |
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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
−13.04913727363512579253475579392, −12.07098254741489874733056527222, −11.16567476976165648613187736505, −9.659122053155969558081958988247, −8.436743048547681304540774827586, −7.67024011723645141427008216088, −6.21876328928315258291377157276, −3.99096818031750137711652044456, −3.19094457076919009030847200644, 0,
3.19094457076919009030847200644, 3.99096818031750137711652044456, 6.21876328928315258291377157276, 7.67024011723645141427008216088, 8.436743048547681304540774827586, 9.659122053155969558081958988247, 11.16567476976165648613187736505, 12.07098254741489874733056527222, 13.04913727363512579253475579392
