| L(s) = 1 | − 1.77·2-s + 9.46·3-s − 28.8·4-s + 25·5-s − 16.7·6-s + 232.·7-s + 108.·8-s − 153.·9-s − 44.3·10-s + 121·11-s − 272.·12-s − 324.·13-s − 412.·14-s + 236.·15-s + 731.·16-s + 1.08e3·17-s + 272.·18-s − 361·19-s − 721.·20-s + 2.20e3·21-s − 214.·22-s − 3.84e3·23-s + 1.02e3·24-s + 625·25-s + 576.·26-s − 3.75e3·27-s − 6.70e3·28-s + ⋯ |
| L(s) = 1 | − 0.313·2-s + 0.607·3-s − 0.901·4-s + 0.447·5-s − 0.190·6-s + 1.79·7-s + 0.596·8-s − 0.631·9-s − 0.140·10-s + 0.301·11-s − 0.547·12-s − 0.533·13-s − 0.562·14-s + 0.271·15-s + 0.714·16-s + 0.912·17-s + 0.198·18-s − 0.229·19-s − 0.403·20-s + 1.08·21-s − 0.0946·22-s − 1.51·23-s + 0.362·24-s + 0.200·25-s + 0.167·26-s − 0.990·27-s − 1.61·28-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 | \( 1 + 1.77T + 32T^{2} \) |
| 3 | \( 1 - 9.46T + 243T^{2} \) |
| 7 | \( 1 - 232.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 324.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.84e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.71e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.83e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.53e4T + 8.58e9T^{2} \) |
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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
−8.600712694159236770185295384630, −8.085346106948721295349125420041, −7.64953824721768448941125146378, −6.05291592413650695917133988680, −5.17547462017885194650771285262, −4.50291477235224281019496822996, −3.43548515081637631297882876232, −2.08254873018691426086204664348, −1.37080731847564323078236480826, 0,
1.37080731847564323078236480826, 2.08254873018691426086204664348, 3.43548515081637631297882876232, 4.50291477235224281019496822996, 5.17547462017885194650771285262, 6.05291592413650695917133988680, 7.64953824721768448941125146378, 8.085346106948721295349125420041, 8.600712694159236770185295384630
