| L(s) = 1 | + 3.56·2-s + 4.68·4-s + 15.6·5-s + 7·7-s − 11.8·8-s + 55.8·10-s − 11·11-s + 52.9·13-s + 24.9·14-s − 79.5·16-s + 77.1·17-s + 13.4·19-s + 73.4·20-s − 39.1·22-s + 59.0·23-s + 121.·25-s + 188.·26-s + 32.7·28-s + 69.3·29-s + 75.9·31-s − 188.·32-s + 274.·34-s + 109.·35-s − 335.·37-s + 47.7·38-s − 185.·40-s + 318.·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.585·4-s + 1.40·5-s + 0.377·7-s − 0.521·8-s + 1.76·10-s − 0.301·11-s + 1.12·13-s + 0.475·14-s − 1.24·16-s + 1.10·17-s + 0.161·19-s + 0.821·20-s − 0.379·22-s + 0.535·23-s + 0.968·25-s + 1.42·26-s + 0.221·28-s + 0.443·29-s + 0.440·31-s − 1.04·32-s + 1.38·34-s + 0.530·35-s − 1.49·37-s + 0.203·38-s − 0.732·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.319264612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.319264612\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.56T + 8T^{2} \) |
| 5 | \( 1 - 15.6T + 125T^{2} \) |
| 13 | \( 1 - 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 59.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 57.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 315.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 598.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 337.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 107.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 405.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 133.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 287.T + 9.12e5T^{2} \) |
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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
−10.17994027949927581836812395125, −9.238183426258638488630926694384, −8.449384439413613906694662761847, −7.10078077422777306636968571432, −5.98695121282127991147654217037, −5.63209837978939296006271690091, −4.71271466931354847951921708772, −3.52784013374602916837571307139, −2.52300906503682964861478311589, −1.24021236054099616543984647204,
1.24021236054099616543984647204, 2.52300906503682964861478311589, 3.52784013374602916837571307139, 4.71271466931354847951921708772, 5.63209837978939296006271690091, 5.98695121282127991147654217037, 7.10078077422777306636968571432, 8.449384439413613906694662761847, 9.238183426258638488630926694384, 10.17994027949927581836812395125
