| L(s) = 1 | − 3.49·2-s + 4.19·4-s + 7.87·5-s + 13.2·8-s − 27.5·10-s − 54.0·11-s − 48.9·13-s − 79.9·16-s − 96.9·17-s − 142.·19-s + 33.0·20-s + 188.·22-s − 103.·23-s − 62.9·25-s + 170.·26-s − 24.5·29-s + 187.·31-s + 172.·32-s + 338.·34-s + 146.·37-s + 497.·38-s + 104.·40-s − 314.·41-s + 173.·43-s − 226.·44-s + 362.·46-s + 259.·47-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.524·4-s + 0.704·5-s + 0.587·8-s − 0.869·10-s − 1.48·11-s − 1.04·13-s − 1.24·16-s − 1.38·17-s − 1.71·19-s + 0.369·20-s + 1.82·22-s − 0.940·23-s − 0.503·25-s + 1.28·26-s − 0.157·29-s + 1.08·31-s + 0.955·32-s + 1.70·34-s + 0.651·37-s + 2.12·38-s + 0.413·40-s − 1.19·41-s + 0.614·43-s − 0.776·44-s + 1.16·46-s + 0.804·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3562844500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3562844500\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.49T + 8T^{2} \) |
| 5 | \( 1 - 7.87T + 125T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 314.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 259.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 620.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 443.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 113.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 41.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 447.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 434.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 329.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 24.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 499.T + 9.12e5T^{2} \) |
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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
−9.372945560846556579223907623160, −8.374769461188760483046962842362, −7.990189338735132467873071635436, −6.97543623254963569369277121786, −6.16797279585251078814860166172, −5.02427307010313947474787082723, −4.28088690482914133229388945107, −2.38256583256639514564599558328, −2.07839319178881540484981670390, −0.33833020353531410404616194610,
0.33833020353531410404616194610, 2.07839319178881540484981670390, 2.38256583256639514564599558328, 4.28088690482914133229388945107, 5.02427307010313947474787082723, 6.16797279585251078814860166172, 6.97543623254963569369277121786, 7.990189338735132467873071635436, 8.374769461188760483046962842362, 9.372945560846556579223907623160
