| L(s) = 1 | + 2·2-s − 1.93·3-s + 4·4-s + 5·5-s − 3.87·6-s + 2.87·7-s + 8·8-s − 23.2·9-s + 10·10-s − 7.75·12-s + 31.3·13-s + 5.74·14-s − 9.69·15-s + 16·16-s − 80.0·17-s − 46.4·18-s − 28.2·19-s + 20·20-s − 5.57·21-s − 11.0·23-s − 15.5·24-s + 25·25-s + 62.7·26-s + 97.4·27-s + 11.4·28-s − 149.·29-s − 19.3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.373·3-s + 0.5·4-s + 0.447·5-s − 0.263·6-s + 0.155·7-s + 0.353·8-s − 0.860·9-s + 0.316·10-s − 0.186·12-s + 0.669·13-s + 0.109·14-s − 0.166·15-s + 0.250·16-s − 1.14·17-s − 0.608·18-s − 0.341·19-s + 0.223·20-s − 0.0578·21-s − 0.100·23-s − 0.131·24-s + 0.200·25-s + 0.473·26-s + 0.694·27-s + 0.0775·28-s − 0.955·29-s − 0.118·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.93T + 27T^{2} \) |
| 7 | \( 1 - 2.87T + 343T^{2} \) |
| 13 | \( 1 - 31.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 11.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.22T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 419.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 239.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 382.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 450.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 485.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3T + 9.12e5T^{2} \) |
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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
−8.857808935615697711226689994817, −8.215717706336029434234318129740, −6.98103646688610669922122225649, −6.27341894685483511023529723730, −5.58360150775876041563482971655, −4.76451579749283701591750186336, −3.73303192700166607835239316065, −2.65403126152802991074339498866, −1.61073936490929327909545325067, 0,
1.61073936490929327909545325067, 2.65403126152802991074339498866, 3.73303192700166607835239316065, 4.76451579749283701591750186336, 5.58360150775876041563482971655, 6.27341894685483511023529723730, 6.98103646688610669922122225649, 8.215717706336029434234318129740, 8.857808935615697711226689994817
