| L(s) = 1 | + 3.02·2-s − 27.7·3-s − 22.8·4-s + 25·5-s − 83.9·6-s + 36.4·7-s − 165.·8-s + 527.·9-s + 75.5·10-s − 121·11-s + 634.·12-s − 633.·13-s + 110.·14-s − 694.·15-s + 229.·16-s − 80.9·17-s + 1.59e3·18-s + 361·19-s − 571.·20-s − 1.01e3·21-s − 365.·22-s − 3.32e3·23-s + 4.60e3·24-s + 625·25-s − 1.91e3·26-s − 7.90e3·27-s − 833.·28-s + ⋯ |
| L(s) = 1 | + 0.534·2-s − 1.78·3-s − 0.714·4-s + 0.447·5-s − 0.951·6-s + 0.281·7-s − 0.916·8-s + 2.17·9-s + 0.239·10-s − 0.301·11-s + 1.27·12-s − 1.03·13-s + 0.150·14-s − 0.796·15-s + 0.224·16-s − 0.0679·17-s + 1.16·18-s + 0.229·19-s − 0.319·20-s − 0.501·21-s − 0.161·22-s − 1.30·23-s + 1.63·24-s + 0.200·25-s − 0.555·26-s − 2.08·27-s − 0.200·28-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 | \( 1 - 3.02T + 32T^{2} \) |
| 3 | \( 1 + 27.7T + 243T^{2} \) |
| 7 | \( 1 - 36.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 633.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 80.9T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.88e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.54e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.47e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.71e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.80e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.24e4T + 8.58e9T^{2} \) |
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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
−8.942092905310098717869932417647, −7.72899727169090935051944724366, −6.77917613496740075691386947259, −5.88139953632460144490281185141, −5.31849341491551923639286357249, −4.73216510271760581216103660899, −3.85225889141667221487956609597, −2.24713293472686097420051241180, −0.859697400886470316458930071186, 0,
0.859697400886470316458930071186, 2.24713293472686097420051241180, 3.85225889141667221487956609597, 4.73216510271760581216103660899, 5.31849341491551923639286357249, 5.88139953632460144490281185141, 6.77917613496740075691386947259, 7.72899727169090935051944724366, 8.942092905310098717869932417647
