| L(s) = 1 | + 0.540·2-s − 3·3-s − 7.70·4-s + 5·5-s − 1.62·6-s + 24.4·7-s − 8.49·8-s + 9·9-s + 2.70·10-s − 11·11-s + 23.1·12-s − 84.5·13-s + 13.2·14-s − 15·15-s + 57.0·16-s − 62.8·17-s + 4.86·18-s − 159.·19-s − 38.5·20-s − 73.3·21-s − 5.94·22-s − 114.·23-s + 25.4·24-s + 25·25-s − 45.7·26-s − 27·27-s − 188.·28-s + ⋯ |
| L(s) = 1 | + 0.191·2-s − 0.577·3-s − 0.963·4-s + 0.447·5-s − 0.110·6-s + 1.32·7-s − 0.375·8-s + 0.333·9-s + 0.0854·10-s − 0.301·11-s + 0.556·12-s − 1.80·13-s + 0.252·14-s − 0.258·15-s + 0.891·16-s − 0.895·17-s + 0.0637·18-s − 1.92·19-s − 0.430·20-s − 0.762·21-s − 0.0576·22-s − 1.03·23-s + 0.216·24-s + 0.200·25-s − 0.344·26-s − 0.192·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.540T + 8T^{2} \) |
| 7 | \( 1 - 24.4T + 343T^{2} \) |
| 13 | \( 1 + 84.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.87T + 2.97e4T^{2} \) |
| 37 | \( 1 + 14.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 463.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 486.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 118.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 273.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 884.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 720.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 71.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 146.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 399.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−12.06532263044138090557330186720, −10.78788034835486251756389641174, −9.974342647212632823202518823828, −8.759653361251736844892746679812, −7.79221230379473379016271373658, −6.28360587873121776552872967025, −4.92423063742327970705801464112, −4.51229730165067202985344612662, −2.11135438955377781157363479411, 0,
2.11135438955377781157363479411, 4.51229730165067202985344612662, 4.92423063742327970705801464112, 6.28360587873121776552872967025, 7.79221230379473379016271373658, 8.759653361251736844892746679812, 9.974342647212632823202518823828, 10.78788034835486251756389641174, 12.06532263044138090557330186720
