| L(s) = 1 | − 3.32·2-s + 3.07·4-s + 5·5-s + 12.1·7-s + 16.3·8-s − 16.6·10-s − 11·11-s − 57.2·13-s − 40.3·14-s − 79.1·16-s − 22.7·17-s − 46.2·19-s + 15.3·20-s + 36.6·22-s + 128.·23-s + 25·25-s + 190.·26-s + 37.2·28-s + 71.6·29-s − 88.7·31-s + 132.·32-s + 75.8·34-s + 60.5·35-s − 30.5·37-s + 153.·38-s + 81.9·40-s + 223.·41-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.384·4-s + 0.447·5-s + 0.653·7-s + 0.724·8-s − 0.526·10-s − 0.301·11-s − 1.22·13-s − 0.769·14-s − 1.23·16-s − 0.325·17-s − 0.558·19-s + 0.171·20-s + 0.354·22-s + 1.16·23-s + 0.200·25-s + 1.43·26-s + 0.251·28-s + 0.458·29-s − 0.513·31-s + 0.730·32-s + 0.382·34-s + 0.292·35-s − 0.135·37-s + 0.656·38-s + 0.323·40-s + 0.850·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 3.32T + 8T^{2} \) |
| 7 | \( 1 - 12.1T + 343T^{2} \) |
| 13 | \( 1 + 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 30.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 76.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 258.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 97.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 278.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 482.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 93.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 60.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 662.T + 9.12e5T^{2} \) |
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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
−9.985346931355463503718442569990, −9.194780952119227729848686419073, −8.427026788987430018700984070540, −7.55836639148554784698452398896, −6.75082745479393022265473482527, −5.26064056363853332084021802851, −4.47813594797995885325495488947, −2.58940603114488846480828129517, −1.45901574315025026962437308797, 0,
1.45901574315025026962437308797, 2.58940603114488846480828129517, 4.47813594797995885325495488947, 5.26064056363853332084021802851, 6.75082745479393022265473482527, 7.55836639148554784698452398896, 8.427026788987430018700984070540, 9.194780952119227729848686419073, 9.985346931355463503718442569990
