| L(s) = 1 | − 7.58·2-s + 22.1·3-s + 25.4·4-s − 25·5-s − 168.·6-s + 53.6·7-s + 49.3·8-s + 248.·9-s + 189.·10-s + 121·11-s + 565.·12-s + 540.·13-s − 406.·14-s − 554.·15-s − 1.18e3·16-s + 1.19e3·17-s − 1.88e3·18-s + 361·19-s − 637.·20-s + 1.18e3·21-s − 917.·22-s − 2.76e3·23-s + 1.09e3·24-s + 625·25-s − 4.10e3·26-s + 122.·27-s + 1.36e3·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 1.42·3-s + 0.796·4-s − 0.447·5-s − 1.90·6-s + 0.413·7-s + 0.272·8-s + 1.02·9-s + 0.599·10-s + 0.301·11-s + 1.13·12-s + 0.887·13-s − 0.554·14-s − 0.636·15-s − 1.16·16-s + 1.00·17-s − 1.37·18-s + 0.229·19-s − 0.356·20-s + 0.588·21-s − 0.404·22-s − 1.09·23-s + 0.387·24-s + 0.200·25-s − 1.18·26-s + 0.0323·27-s + 0.329·28-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 | \( 1 + 7.58T + 32T^{2} \) |
| 3 | \( 1 - 22.1T + 243T^{2} \) |
| 7 | \( 1 - 53.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 540.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.19e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 809.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.46e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.45e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.44e4T + 8.58e9T^{2} \) |
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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
−8.550463163096882204258534030742, −8.280116654664703276810574958898, −7.58444471350348861989945411126, −6.78586365373653867044114900219, −5.32553649795523734311891624416, −3.95482986762042313582879795852, −3.34150070973793409432839075667, −1.96602999842872346325936074667, −1.34593684448455972029519235775, 0,
1.34593684448455972029519235775, 1.96602999842872346325936074667, 3.34150070973793409432839075667, 3.95482986762042313582879795852, 5.32553649795523734311891624416, 6.78586365373653867044114900219, 7.58444471350348861989945411126, 8.280116654664703276810574958898, 8.550463163096882204258534030742
