| L(s) = 1 | − 10.5·2-s + 0.539·3-s + 78.6·4-s − 25·5-s − 5.67·6-s + 3.26·7-s − 490.·8-s − 242.·9-s + 262.·10-s − 121·11-s + 42.4·12-s + 723.·13-s − 34.3·14-s − 13.4·15-s + 2.64e3·16-s − 1.17e3·17-s + 2.55e3·18-s − 361·19-s − 1.96e3·20-s + 1.76·21-s + 1.27e3·22-s − 4.11e3·23-s − 265.·24-s + 625·25-s − 7.61e3·26-s − 262.·27-s + 256.·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.0346·3-s + 2.45·4-s − 0.447·5-s − 0.0643·6-s + 0.0251·7-s − 2.71·8-s − 0.998·9-s + 0.831·10-s − 0.301·11-s + 0.0851·12-s + 1.18·13-s − 0.0467·14-s − 0.0154·15-s + 2.58·16-s − 0.986·17-s + 1.85·18-s − 0.229·19-s − 1.09·20-s + 0.000871·21-s + 0.560·22-s − 1.62·23-s − 0.0939·24-s + 0.200·25-s − 2.20·26-s − 0.0692·27-s + 0.0618·28-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 | \( 1 + 10.5T + 32T^{2} \) |
| 3 | \( 1 - 0.539T + 243T^{2} \) |
| 7 | \( 1 - 3.26T + 1.68e4T^{2} \) |
| 13 | \( 1 - 723.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.17e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.59e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.96e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.07e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.21e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.58e4T + 8.58e9T^{2} \) |
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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
−8.549288211431731801403006118127, −8.313306023043150674202593524210, −7.48689530661248338821280465588, −6.39081974887100746061830006868, −5.90492470507722614335144899094, −4.20278046174516151350252351774, −2.92265330869630878148963183989, −2.08254614075567550231903155368, −0.857839998959105373249501586931, 0,
0.857839998959105373249501586931, 2.08254614075567550231903155368, 2.92265330869630878148963183989, 4.20278046174516151350252351774, 5.90492470507722614335144899094, 6.39081974887100746061830006868, 7.48689530661248338821280465588, 8.313306023043150674202593524210, 8.549288211431731801403006118127
