| L(s) = 1 | + 8.57·2-s + 16.0·3-s + 41.4·4-s + 25·5-s + 137.·6-s + 3.71·7-s + 81.1·8-s + 14.4·9-s + 214.·10-s − 121·11-s + 665.·12-s + 94.0·13-s + 31.8·14-s + 401.·15-s − 631.·16-s + 196.·17-s + 124.·18-s − 425.·19-s + 1.03e3·20-s + 59.6·21-s − 1.03e3·22-s + 518.·23-s + 1.30e3·24-s + 625·25-s + 805.·26-s − 3.66e3·27-s + 154.·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.02·3-s + 1.29·4-s + 0.447·5-s + 1.55·6-s + 0.0286·7-s + 0.448·8-s + 0.0596·9-s + 0.677·10-s − 0.301·11-s + 1.33·12-s + 0.154·13-s + 0.0434·14-s + 0.460·15-s − 0.616·16-s + 0.165·17-s + 0.0903·18-s − 0.270·19-s + 0.579·20-s + 0.0294·21-s − 0.456·22-s + 0.204·23-s + 0.461·24-s + 0.200·25-s + 0.233·26-s − 0.968·27-s + 0.0371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.844282077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.844282077\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 121T \) |
| good | 2 | \( 1 - 8.57T + 32T^{2} \) |
| 3 | \( 1 - 16.0T + 243T^{2} \) |
| 7 | \( 1 - 3.71T + 1.68e4T^{2} \) |
| 13 | \( 1 - 94.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 196.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 425.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 518.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 378.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.45e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.48e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.41e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 551.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.29e4T + 8.58e9T^{2} \) |
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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
−14.16274416991818887723589027144, −13.45113922903597237652156448746, −12.51190890843967129273307445417, −11.15033518551979842767683092876, −9.514230490183577453994395600947, −8.169560904971295204726155912779, −6.47617545963532347570851984486, −5.10955704809164212867502394472, −3.55952998132502126454546309691, −2.36224490516250573872248365349,
2.36224490516250573872248365349, 3.55952998132502126454546309691, 5.10955704809164212867502394472, 6.47617545963532347570851984486, 8.169560904971295204726155912779, 9.514230490183577453994395600947, 11.15033518551979842767683092876, 12.51190890843967129273307445417, 13.45113922903597237652156448746, 14.16274416991818887723589027144
