L(s) = 1 | − 4.54·2-s − 4.48·3-s − 11.3·4-s − 25·5-s + 20.3·6-s + 16.9·7-s + 196.·8-s − 222.·9-s + 113.·10-s − 121·11-s + 50.7·12-s + 146.·13-s − 77.0·14-s + 112.·15-s − 533.·16-s − 1.03e3·17-s + 1.01e3·18-s + 361·19-s + 283.·20-s − 76.0·21-s + 550.·22-s + 754.·23-s − 883.·24-s + 625·25-s − 666.·26-s + 2.08e3·27-s − 192.·28-s + ⋯ |
L(s) = 1 | − 0.803·2-s − 0.287·3-s − 0.353·4-s − 0.447·5-s + 0.231·6-s + 0.130·7-s + 1.08·8-s − 0.917·9-s + 0.359·10-s − 0.301·11-s + 0.101·12-s + 0.240·13-s − 0.105·14-s + 0.128·15-s − 0.520·16-s − 0.870·17-s + 0.737·18-s + 0.229·19-s + 0.158·20-s − 0.0376·21-s + 0.242·22-s + 0.297·23-s − 0.313·24-s + 0.200·25-s − 0.193·26-s + 0.551·27-s − 0.0462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.3161468857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3161468857\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 4.54T + 32T^{2} \) |
| 3 | \( 1 + 4.48T + 243T^{2} \) |
| 7 | \( 1 - 16.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 146.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.03e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 754.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 865.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.71e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.61e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.17e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.06e4T + 8.58e9T^{2} \) |
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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
−8.968801495942828138020043956047, −8.550469309675808019862412736700, −7.71969841192811716280517369663, −6.89342071911407105641244981253, −5.73627741343012773640385561320, −4.89380021300069883339554707674, −3.98248108149713437439015467866, −2.78777737991820633147243144995, −1.46332954435355957613899209597, −0.29296603006653273850234268433,
0.29296603006653273850234268433, 1.46332954435355957613899209597, 2.78777737991820633147243144995, 3.98248108149713437439015467866, 4.89380021300069883339554707674, 5.73627741343012773640385561320, 6.89342071911407105641244981253, 7.71969841192811716280517369663, 8.550469309675808019862412736700, 8.968801495942828138020043956047