| 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
−8.341005517370873595726996050605, −8.008016446225899963198630793391, −7.51024253202192400671059025965, −5.79881176367497587685502491560, −5.11365212023176254933923598621, −4.31042036550959526311334692016, −3.44448883571955601531361825942, −2.27789840407385858825632378030, −1.61961822146114608508061610694, 0,
1.61961822146114608508061610694, 2.27789840407385858825632378030, 3.44448883571955601531361825942, 4.31042036550959526311334692016, 5.11365212023176254933923598621, 5.79881176367497587685502491560, 7.51024253202192400671059025965, 8.008016446225899963198630793391, 8.341005517370873595726996050605
