| L(s) = 1 | − 25·5-s + 221.·7-s + 364.·11-s − 888.·13-s − 290.·17-s − 146.·19-s − 1.80e3·23-s + 625·25-s − 4.30e3·29-s − 2.42e3·31-s − 5.53e3·35-s + 7.41e3·37-s − 1.56e4·41-s − 483.·43-s − 1.91e4·47-s + 3.21e4·49-s + 1.29e4·53-s − 9.11e3·55-s + 3.17e4·59-s − 1.70e4·61-s + 2.22e4·65-s + 3.78e3·67-s − 1.93e4·71-s + 1.08e4·73-s + 8.06e4·77-s + 1.75e3·79-s + 9.72e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.70·7-s + 0.908·11-s − 1.45·13-s − 0.243·17-s − 0.0930·19-s − 0.712·23-s + 0.200·25-s − 0.950·29-s − 0.453·31-s − 0.763·35-s + 0.890·37-s − 1.45·41-s − 0.0398·43-s − 1.26·47-s + 1.91·49-s + 0.630·53-s − 0.406·55-s + 1.18·59-s − 0.586·61-s + 0.652·65-s + 0.103·67-s − 0.456·71-s + 0.238·73-s + 1.55·77-s + 0.0315·79-s + 1.54·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 364.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 888.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 290.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 146.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.56e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 483.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.78e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.75e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.35e5T + 8.58e9T^{2} \) |
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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
−9.153811151774606505484070760533, −8.225151185362749193068137370356, −7.60852046568073659660074358964, −6.74857282805641633585523368177, −5.40151613951991936026258905856, −4.66048942989297777520182108996, −3.83197455839453872463106110804, −2.29944843159127460671473166947, −1.41818797794965562158541894731, 0,
1.41818797794965562158541894731, 2.29944843159127460671473166947, 3.83197455839453872463106110804, 4.66048942989297777520182108996, 5.40151613951991936026258905856, 6.74857282805641633585523368177, 7.60852046568073659660074358964, 8.225151185362749193068137370356, 9.153811151774606505484070760533
