| L(s) = 1 | − 4·2-s + 16·4-s − 59.5·5-s − 64·8-s + 238.·10-s + 616.·11-s − 418.·13-s + 256·16-s + 1.79e3·17-s − 1.27e3·19-s − 953.·20-s − 2.46e3·22-s − 4.78e3·23-s + 426.·25-s + 1.67e3·26-s − 1.71e3·29-s − 642.·31-s − 1.02e3·32-s − 7.17e3·34-s − 2.36e3·37-s + 5.11e3·38-s + 3.81e3·40-s + 1.56e4·41-s − 1.63e3·43-s + 9.86e3·44-s + 1.91e4·46-s + 2.07e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.06·5-s − 0.353·8-s + 0.753·10-s + 1.53·11-s − 0.686·13-s + 0.250·16-s + 1.50·17-s − 0.813·19-s − 0.533·20-s − 1.08·22-s − 1.88·23-s + 0.136·25-s + 0.485·26-s − 0.378·29-s − 0.120·31-s − 0.176·32-s − 1.06·34-s − 0.283·37-s + 0.575·38-s + 0.376·40-s + 1.45·41-s − 0.135·43-s + 0.767·44-s + 1.33·46-s + 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 59.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 616.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 418.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.79e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 642.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.56e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.27e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.43e4T + 8.58e9T^{2} \) |
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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
−8.991401496919358282146114694328, −8.002081889761395316410637999759, −7.55430958542966741602841448909, −6.57017348092223294919284127802, −5.66512933122514901580679591304, −4.16936035130082708753465584189, −3.65836644967760464957873419020, −2.22497586143914281034170014326, −1.04420409080615409248609403174, 0,
1.04420409080615409248609403174, 2.22497586143914281034170014326, 3.65836644967760464957873419020, 4.16936035130082708753465584189, 5.66512933122514901580679591304, 6.57017348092223294919284127802, 7.55430958542966741602841448909, 8.002081889761395316410637999759, 8.991401496919358282146114694328
