| L(s) = 1 | + 4.13·2-s − 5.83·3-s + 9.12·4-s − 0.336·5-s − 24.1·6-s + 10.5·7-s + 4.64·8-s + 7.06·9-s − 1.39·10-s − 53.2·12-s + 13·13-s + 43.5·14-s + 1.96·15-s − 53.7·16-s + 28.0·17-s + 29.2·18-s − 2.19·19-s − 3.07·20-s − 61.4·21-s + 188.·23-s − 27.1·24-s − 124.·25-s + 53.7·26-s + 116.·27-s + 95.9·28-s − 69.4·29-s + 8.12·30-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 1.12·3-s + 1.14·4-s − 0.0301·5-s − 1.64·6-s + 0.568·7-s + 0.205·8-s + 0.261·9-s − 0.0440·10-s − 1.28·12-s + 0.277·13-s + 0.831·14-s + 0.0338·15-s − 0.839·16-s + 0.400·17-s + 0.382·18-s − 0.0264·19-s − 0.0343·20-s − 0.638·21-s + 1.71·23-s − 0.230·24-s − 0.999·25-s + 0.405·26-s + 0.829·27-s + 0.647·28-s − 0.444·29-s + 0.0494·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 4.13T + 8T^{2} \) |
| 3 | \( 1 + 5.83T + 27T^{2} \) |
| 5 | \( 1 + 0.336T + 125T^{2} \) |
| 7 | \( 1 - 10.5T + 343T^{2} \) |
| 17 | \( 1 - 28.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.19T + 6.85e3T^{2} \) |
| 23 | \( 1 - 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 110.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 264.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 81.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 275.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 650.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 719.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 56.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 426.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 412.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 910.T + 9.12e5T^{2} \) |
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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
−8.638606534474573033292772179579, −7.49679280087194992500830181896, −6.69140269526900818730681212657, −5.85204808798781205825728787631, −5.34495476167974335172722460819, −4.69419479032542366675010859742, −3.77717951411812477023084061687, −2.79315791869474533196932794216, −1.42749911177762369193378349751, 0,
1.42749911177762369193378349751, 2.79315791869474533196932794216, 3.77717951411812477023084061687, 4.69419479032542366675010859742, 5.34495476167974335172722460819, 5.85204808798781205825728787631, 6.69140269526900818730681212657, 7.49679280087194992500830181896, 8.638606534474573033292772179579
