| L(s) = 1 | + 0.213·2-s − 7.23·3-s − 7.95·4-s − 3.30·5-s − 1.54·6-s − 30.5·7-s − 3.41·8-s + 25.3·9-s − 0.706·10-s − 66.6·11-s + 57.5·12-s + 32.7·13-s − 6.53·14-s + 23.9·15-s + 62.9·16-s − 58.7·17-s + 5.42·18-s − 53.7·19-s + 26.2·20-s + 221.·21-s − 14.2·22-s − 169.·23-s + 24.6·24-s − 114.·25-s + 7.00·26-s + 11.9·27-s + 243.·28-s + ⋯ |
| L(s) = 1 | + 0.0756·2-s − 1.39·3-s − 0.994·4-s − 0.295·5-s − 0.105·6-s − 1.65·7-s − 0.150·8-s + 0.938·9-s − 0.0223·10-s − 1.82·11-s + 1.38·12-s + 0.698·13-s − 0.124·14-s + 0.411·15-s + 0.982·16-s − 0.837·17-s + 0.0709·18-s − 0.649·19-s + 0.294·20-s + 2.29·21-s − 0.138·22-s − 1.53·23-s + 0.209·24-s − 0.912·25-s + 0.0528·26-s + 0.0849·27-s + 1.64·28-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 | \( 1 - 0.213T + 8T^{2} \) |
| 3 | \( 1 + 7.23T + 27T^{2} \) |
| 5 | \( 1 + 3.30T + 125T^{2} \) |
| 7 | \( 1 + 30.5T + 343T^{2} \) |
| 11 | \( 1 + 66.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 67.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.72T + 2.97e4T^{2} \) |
| 37 | \( 1 - 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 24.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 597.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 205.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 392.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 607.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 98.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 874.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 788.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 495.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 528.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 2.16T + 9.12e5T^{2} \) |
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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
−8.414495525456894696415083720783, −7.78303691284686689451349788718, −6.50501779104856659102634659501, −6.05896053504856671604954612334, −5.37203176652064600768388699156, −4.43832092758452739387053049589, −3.68960735508187602624123114093, −2.51271662277502356924067413676, −0.54030699509634702015587203526, 0,
0.54030699509634702015587203526, 2.51271662277502356924067413676, 3.68960735508187602624123114093, 4.43832092758452739387053049589, 5.37203176652064600768388699156, 6.05896053504856671604954612334, 6.50501779104856659102634659501, 7.78303691284686689451349788718, 8.414495525456894696415083720783
