L(s) = 1 | + 3.20·2-s + 9·3-s − 21.7·4-s + 26.7·5-s + 28.8·6-s + 39.0·7-s − 172.·8-s + 81·9-s + 85.6·10-s − 606.·11-s − 195.·12-s − 161.·13-s + 125.·14-s + 240.·15-s + 142.·16-s − 1.65e3·17-s + 259.·18-s + 882.·19-s − 580.·20-s + 351.·21-s − 1.94e3·22-s + 2.92e3·23-s − 1.55e3·24-s − 2.41e3·25-s − 518.·26-s + 729·27-s − 847.·28-s + ⋯ |
L(s) = 1 | + 0.566·2-s + 0.577·3-s − 0.678·4-s + 0.478·5-s + 0.327·6-s + 0.301·7-s − 0.951·8-s + 0.333·9-s + 0.270·10-s − 1.51·11-s − 0.391·12-s − 0.265·13-s + 0.170·14-s + 0.276·15-s + 0.139·16-s − 1.38·17-s + 0.188·18-s + 0.560·19-s − 0.324·20-s + 0.173·21-s − 0.856·22-s + 1.15·23-s − 0.549·24-s − 0.771·25-s − 0.150·26-s + 0.192·27-s − 0.204·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 - 3.20T + 32T^{2} \) |
| 5 | \( 1 - 26.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 39.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 606.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 161.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.65e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 882.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.92e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.67e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.50e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.10e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 3.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 26.9T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.66e4T + 8.58e9T^{2} \) |
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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
−11.35430046687357032202364537781, −10.17457709440697623104184896636, −9.221183020232604127936133190042, −8.348289444582176558661839750272, −7.15958189318662469081587646001, −5.55433893502766979357090847529, −4.79641795752196435130653842974, −3.39295175934334592423867425948, −2.11915960750502431681863590638, 0,
2.11915960750502431681863590638, 3.39295175934334592423867425948, 4.79641795752196435130653842974, 5.55433893502766979357090847529, 7.15958189318662469081587646001, 8.348289444582176558661839750272, 9.221183020232604127936133190042, 10.17457709440697623104184896636, 11.35430046687357032202364537781