| L(s) = 1 | − 0.501·2-s − 9·3-s − 31.7·4-s − 14.0·5-s + 4.51·6-s + 117.·7-s + 31.9·8-s + 81·9-s + 7.04·10-s + 53.5·11-s + 285.·12-s − 471.·13-s − 58.8·14-s + 126.·15-s + 9.99e2·16-s + 1.16e3·17-s − 40.6·18-s + 1.82e3·19-s + 446.·20-s − 1.05e3·21-s − 26.8·22-s + 254.·23-s − 287.·24-s − 2.92e3·25-s + 236.·26-s − 729·27-s − 3.72e3·28-s + ⋯ |
| L(s) = 1 | − 0.0886·2-s − 0.577·3-s − 0.992·4-s − 0.251·5-s + 0.0512·6-s + 0.904·7-s + 0.176·8-s + 0.333·9-s + 0.0222·10-s + 0.133·11-s + 0.572·12-s − 0.774·13-s − 0.0801·14-s + 0.145·15-s + 0.976·16-s + 0.977·17-s − 0.0295·18-s + 1.16·19-s + 0.249·20-s − 0.522·21-s − 0.0118·22-s + 0.100·23-s − 0.102·24-s − 0.936·25-s + 0.0686·26-s − 0.192·27-s − 0.897·28-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 | \( 1 + 0.501T + 32T^{2} \) |
| 5 | \( 1 + 14.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 117.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 53.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 471.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.82e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 254.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.74e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.85e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.66e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 3.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e4T + 8.58e9T^{2} \) |
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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
−11.47250057806313265548177668321, −10.18695290815484730145457425979, −9.413756950308186690040968181818, −8.118700587364579361914324142701, −7.35576666733020258045283293496, −5.56992484267077749937967424986, −4.88636311218501977113863508411, −3.62407186945185970731829578793, −1.42824967717719035529577203089, 0,
1.42824967717719035529577203089, 3.62407186945185970731829578793, 4.88636311218501977113863508411, 5.56992484267077749937967424986, 7.35576666733020258045283293496, 8.118700587364579361914324142701, 9.413756950308186690040968181818, 10.18695290815484730145457425979, 11.47250057806313265548177668321
