| L(s) = 1 | − 3.82·2-s + 7.76·3-s + 6.61·4-s − 18.1·5-s − 29.6·6-s − 5.19·7-s + 5.28·8-s + 33.2·9-s + 69.3·10-s + 31.5·11-s + 51.3·12-s − 71.3·13-s + 19.8·14-s − 140.·15-s − 73.1·16-s + 136.·17-s − 127.·18-s + 1.39·19-s − 120.·20-s − 40.3·21-s − 120.·22-s − 60.7·23-s + 41.0·24-s + 203.·25-s + 272.·26-s + 48.9·27-s − 34.4·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 1.49·3-s + 0.827·4-s − 1.62·5-s − 2.02·6-s − 0.280·7-s + 0.233·8-s + 1.23·9-s + 2.19·10-s + 0.864·11-s + 1.23·12-s − 1.52·13-s + 0.379·14-s − 2.42·15-s − 1.14·16-s + 1.94·17-s − 1.66·18-s + 0.0169·19-s − 1.34·20-s − 0.419·21-s − 1.16·22-s − 0.550·23-s + 0.348·24-s + 1.63·25-s + 2.05·26-s + 0.348·27-s − 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 3 | \( 1 - 7.76T + 27T^{2} \) |
| 5 | \( 1 + 18.1T + 125T^{2} \) |
| 7 | \( 1 + 5.19T + 343T^{2} \) |
| 11 | \( 1 - 31.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 1.39T + 6.85e3T^{2} \) |
| 23 | \( 1 + 60.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 29.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 87.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 255.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 23.1T + 6.89e4T^{2} \) |
| 47 | \( 1 + 1.08T + 1.03e5T^{2} \) |
| 53 | \( 1 + 66.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 465.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 580.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 328.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 832.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 808.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 875.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 88.7T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.351337809270326243777496608251, −7.930291651512509681515062832553, −7.44661836031158787935323648618, −6.79815483316973815765670490885, −5.00607271754747853361564167957, −3.92416328493174481747849487512, −3.38308068721843571954594753583, −2.31518980448678580602863981382, −1.09924946526129491617409657355, 0,
1.09924946526129491617409657355, 2.31518980448678580602863981382, 3.38308068721843571954594753583, 3.92416328493174481747849487512, 5.00607271754747853361564167957, 6.79815483316973815765670490885, 7.44661836031158787935323648618, 7.930291651512509681515062832553, 8.351337809270326243777496608251
