| L(s) = 1 | − 1.61·2-s + 3·3-s − 5.40·4-s − 4.04·5-s − 4.83·6-s + 7·7-s + 21.5·8-s + 9·9-s + 6.51·10-s − 37.9·11-s − 16.2·12-s + 36.7·13-s − 11.2·14-s − 12.1·15-s + 8.45·16-s − 71.2·17-s − 14.4·18-s + 37.1·19-s + 21.8·20-s + 21·21-s + 61.0·22-s + 23·23-s + 64.7·24-s − 108.·25-s − 59.1·26-s + 27·27-s − 37.8·28-s + ⋯ |
| L(s) = 1 | − 0.569·2-s + 0.577·3-s − 0.675·4-s − 0.361·5-s − 0.328·6-s + 0.377·7-s + 0.954·8-s + 0.333·9-s + 0.206·10-s − 1.03·11-s − 0.390·12-s + 0.783·13-s − 0.215·14-s − 0.208·15-s + 0.132·16-s − 1.01·17-s − 0.189·18-s + 0.448·19-s + 0.244·20-s + 0.218·21-s + 0.591·22-s + 0.208·23-s + 0.550·24-s − 0.869·25-s − 0.446·26-s + 0.192·27-s − 0.255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 1.61T + 8T^{2} \) |
| 5 | \( 1 + 4.04T + 125T^{2} \) |
| 11 | \( 1 + 37.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 296.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 51.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 540.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 40.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 600.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 422.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 414.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 191.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 552.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 659.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 883.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 252.T + 9.12e5T^{2} \) |
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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
−10.06755021085894873872688901167, −9.071265869643276343783973579802, −8.258655662820083560370233750754, −7.88736161319746869411906889407, −6.63324742115102857775248826092, −5.11905518485824741256583165106, −4.31091464667014319765030106983, −3.06319052154090807658911902678, −1.52253523080452762684869148932, 0,
1.52253523080452762684869148932, 3.06319052154090807658911902678, 4.31091464667014319765030106983, 5.11905518485824741256583165106, 6.63324742115102857775248826092, 7.88736161319746869411906889407, 8.258655662820083560370233750754, 9.071265869643276343783973579802, 10.06755021085894873872688901167
