| L(s) = 1 | + 5.56·2-s − 4.22·3-s + 22.9·4-s − 5·5-s − 23.4·6-s − 32.5·7-s + 82.9·8-s − 9.15·9-s − 27.8·10-s + 11·11-s − 96.8·12-s + 52.9·13-s − 180.·14-s + 21.1·15-s + 278.·16-s + 27.1·17-s − 50.8·18-s + 19·19-s − 114.·20-s + 137.·21-s + 61.1·22-s + 165.·23-s − 350.·24-s + 25·25-s + 294.·26-s + 152.·27-s − 745.·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.813·3-s + 2.86·4-s − 0.447·5-s − 1.59·6-s − 1.75·7-s + 3.66·8-s − 0.338·9-s − 0.879·10-s + 0.301·11-s − 2.32·12-s + 1.13·13-s − 3.45·14-s + 0.363·15-s + 4.34·16-s + 0.387·17-s − 0.666·18-s + 0.229·19-s − 1.28·20-s + 1.42·21-s + 0.592·22-s + 1.50·23-s − 2.98·24-s + 0.200·25-s + 2.22·26-s + 1.08·27-s − 5.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.978115405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.978115405\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 5.56T + 8T^{2} \) |
| 3 | \( 1 + 4.22T + 27T^{2} \) |
| 7 | \( 1 + 32.5T + 343T^{2} \) |
| 13 | \( 1 - 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 24.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.85T + 5.06e4T^{2} \) |
| 41 | \( 1 + 66.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 559.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 52.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 702.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 884.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−9.925587413181523899448744179241, −8.607419242766664079548201360747, −7.10066979856496726957654425619, −6.74654191235124366549964481951, −5.82931644381757592071327476906, −5.45805174077666319157315742607, −4.15845006063896907359130614159, −3.44420272608866122534415855035, −2.76131914141805708381063461823, −0.925436329740782174304268672732,
0.925436329740782174304268672732, 2.76131914141805708381063461823, 3.44420272608866122534415855035, 4.15845006063896907359130614159, 5.45805174077666319157315742607, 5.82931644381757592071327476906, 6.74654191235124366549964481951, 7.10066979856496726957654425619, 8.607419242766664079548201360747, 9.925587413181523899448744179241
