| L(s) = 1 | − 2·2-s + 4·4-s − 11.1·5-s + 18.3·7-s − 8·8-s + 22.3·10-s + 23.5·11-s − 67.7·13-s − 36.7·14-s + 16·16-s − 117.·17-s + 110.·19-s − 44.7·20-s − 47.1·22-s − 69.2·23-s + 0.353·25-s + 135.·26-s + 73.5·28-s − 198.·29-s − 311.·31-s − 32·32-s + 234.·34-s − 205.·35-s − 206.·37-s − 220.·38-s + 89.5·40-s + 132.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.00·5-s + 0.993·7-s − 0.353·8-s + 0.708·10-s + 0.646·11-s − 1.44·13-s − 0.702·14-s + 0.250·16-s − 1.67·17-s + 1.33·19-s − 0.500·20-s − 0.456·22-s − 0.627·23-s + 0.00283·25-s + 1.02·26-s + 0.496·28-s − 1.27·29-s − 1.80·31-s − 0.176·32-s + 1.18·34-s − 0.994·35-s − 0.918·37-s − 0.941·38-s + 0.354·40-s + 0.505·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 11.1T + 125T^{2} \) |
| 7 | \( 1 - 18.3T + 343T^{2} \) |
| 11 | \( 1 - 23.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 311.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 335.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 337.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 277.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 665.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 73.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 179.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 846.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 672.T + 9.12e5T^{2} \) |
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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
−11.55147484404414368394701485099, −11.18352807579592093243217712706, −9.731438335001040107315040395565, −8.771067104092293052342967882118, −7.67273223137470657373013128266, −7.05632802395289902446835256584, −5.21620566087277160546712159763, −3.89644642041884438646218258806, −1.98043388523693271977403590444, 0,
1.98043388523693271977403590444, 3.89644642041884438646218258806, 5.21620566087277160546712159763, 7.05632802395289902446835256584, 7.67273223137470657373013128266, 8.771067104092293052342967882118, 9.731438335001040107315040395565, 11.18352807579592093243217712706, 11.55147484404414368394701485099
