| L(s) = 1 | − 9.21·2-s − 14.8·3-s + 52.8·4-s + 25·5-s + 136.·6-s + 144.·7-s − 192.·8-s − 22.9·9-s − 230.·10-s + 121·11-s − 784.·12-s − 526.·13-s − 1.32e3·14-s − 370.·15-s + 80.1·16-s + 1.31e3·17-s + 211.·18-s − 361·19-s + 1.32e3·20-s − 2.13e3·21-s − 1.11e3·22-s + 4.41e3·23-s + 2.85e3·24-s + 625·25-s + 4.84e3·26-s + 3.94e3·27-s + 7.62e3·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.951·3-s + 1.65·4-s + 0.447·5-s + 1.54·6-s + 1.11·7-s − 1.06·8-s − 0.0944·9-s − 0.728·10-s + 0.301·11-s − 1.57·12-s − 0.863·13-s − 1.81·14-s − 0.425·15-s + 0.0782·16-s + 1.10·17-s + 0.153·18-s − 0.229·19-s + 0.739·20-s − 1.05·21-s − 0.491·22-s + 1.74·23-s + 1.01·24-s + 0.200·25-s + 1.40·26-s + 1.04·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 9.21T + 32T^{2} \) |
| 3 | \( 1 + 14.8T + 243T^{2} \) |
| 7 | \( 1 - 144.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 526.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.31e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.61e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.61e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.99e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.96e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.90e4T + 8.58e9T^{2} \) |
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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
−8.631687614803221388014480773736, −8.241565005741371666556753906920, −7.13521060155093618891020288532, −6.59458298919824620837792316789, −5.32589982255295626326201464491, −4.86032447367165527675849478615, −2.97484826321342444512552146026, −1.73132661066437584717295961179, −1.04125893333608956160394861331, 0,
1.04125893333608956160394861331, 1.73132661066437584717295961179, 2.97484826321342444512552146026, 4.86032447367165527675849478615, 5.32589982255295626326201464491, 6.59458298919824620837792316789, 7.13521060155093618891020288532, 8.241565005741371666556753906920, 8.631687614803221388014480773736
