| L(s) = 1 | + 4.70·2-s + 14.1·4-s + 0.830·5-s + 29.0·8-s + 3.90·10-s − 68.2·11-s − 35.7·13-s + 23.2·16-s + 14.6·17-s + 86.7·19-s + 11.7·20-s − 321.·22-s + 58.4·23-s − 124.·25-s − 168.·26-s − 67.1·29-s − 69.5·31-s − 122.·32-s + 69.0·34-s − 289.·37-s + 408.·38-s + 24.0·40-s − 357.·41-s − 235.·43-s − 966.·44-s + 274.·46-s + 450.·47-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.77·4-s + 0.0742·5-s + 1.28·8-s + 0.123·10-s − 1.87·11-s − 0.763·13-s + 0.363·16-s + 0.209·17-s + 1.04·19-s + 0.131·20-s − 3.11·22-s + 0.529·23-s − 0.994·25-s − 1.27·26-s − 0.429·29-s − 0.402·31-s − 0.677·32-s + 0.348·34-s − 1.28·37-s + 1.74·38-s + 0.0952·40-s − 1.36·41-s − 0.836·43-s − 3.31·44-s + 0.881·46-s + 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.70T + 8T^{2} \) |
| 5 | \( 1 - 0.830T + 125T^{2} \) |
| 11 | \( 1 + 68.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 67.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 289.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 357.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 235.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 476.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 804.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 353.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 778.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 74.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.63e3T + 9.12e5T^{2} \) |
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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
−8.784800602037477732774866648862, −7.55947757699532819686043885265, −7.22203596626148400109804093985, −5.95392093044171389644728645990, −5.26294724029638900132206161947, −4.85451331151224034712813048845, −3.59123432968280780572248451908, −2.86373177777671029302926768517, −1.95461313810555395240566264092, 0,
1.95461313810555395240566264092, 2.86373177777671029302926768517, 3.59123432968280780572248451908, 4.85451331151224034712813048845, 5.26294724029638900132206161947, 5.95392093044171389644728645990, 7.22203596626148400109804093985, 7.55947757699532819686043885265, 8.784800602037477732774866648862
