| L(s) = 1 | + 5.28·2-s + 0.757·3-s + 19.9·4-s + 5·5-s + 4.00·6-s + 14.7·7-s + 63.1·8-s − 26.4·9-s + 26.4·10-s + 11·11-s + 15.1·12-s − 8.30·13-s + 77.8·14-s + 3.78·15-s + 174.·16-s + 101.·17-s − 139.·18-s − 19·19-s + 99.7·20-s + 11.1·21-s + 58.1·22-s − 1.36·23-s + 47.8·24-s + 25·25-s − 43.9·26-s − 40.4·27-s + 293.·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.145·3-s + 2.49·4-s + 0.447·5-s + 0.272·6-s + 0.794·7-s + 2.79·8-s − 0.978·9-s + 0.835·10-s + 0.301·11-s + 0.363·12-s − 0.177·13-s + 1.48·14-s + 0.0652·15-s + 2.72·16-s + 1.45·17-s − 1.82·18-s − 0.229·19-s + 1.11·20-s + 0.115·21-s + 0.563·22-s − 0.0123·23-s + 0.407·24-s + 0.200·25-s − 0.331·26-s − 0.288·27-s + 1.98·28-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
| \(L(2)\) |
\(\approx\) |
\(9.095331480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.095331480\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.28T + 8T^{2} \) |
| 3 | \( 1 - 0.757T + 27T^{2} \) |
| 7 | \( 1 - 14.7T + 343T^{2} \) |
| 13 | \( 1 + 8.30T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 1.36T + 1.21e4T^{2} \) |
| 29 | \( 1 - 48.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 0.598T + 5.06e4T^{2} \) |
| 41 | \( 1 + 56.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 35.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 93.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 205.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 607.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 223.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 381.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 755.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 367.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.47e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 651.T + 9.12e5T^{2} \) |
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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
−9.764592558170833121727301345112, −8.448628940302676840820516826145, −7.67460476803819492704562795148, −6.63661292352262556384350432351, −5.80271341128533370877153343650, −5.23900388912548761814901574035, −4.36734367313200610129797211473, −3.29595404922351755845628867159, −2.51903125189932768102343972020, −1.38894025049735117725236280131,
1.38894025049735117725236280131, 2.51903125189932768102343972020, 3.29595404922351755845628867159, 4.36734367313200610129797211473, 5.23900388912548761814901574035, 5.80271341128533370877153343650, 6.63661292352262556384350432351, 7.67460476803819492704562795148, 8.448628940302676840820516826145, 9.764592558170833121727301345112
