| L(s) = 1 | + 10.7·2-s − 9·3-s + 83.9·4-s − 60.5·5-s − 96.8·6-s − 233.·7-s + 558.·8-s + 81·9-s − 652.·10-s + 441.·11-s − 755.·12-s − 615.·13-s − 2.51e3·14-s + 545.·15-s + 3.33e3·16-s − 1.49e3·17-s + 872.·18-s − 1.18e3·19-s − 5.08e3·20-s + 2.09e3·21-s + 4.75e3·22-s − 4.79e3·23-s − 5.02e3·24-s + 546.·25-s − 6.62e3·26-s − 729·27-s − 1.95e4·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.62·4-s − 1.08·5-s − 1.09·6-s − 1.79·7-s + 3.08·8-s + 0.333·9-s − 2.06·10-s + 1.10·11-s − 1.51·12-s − 1.01·13-s − 3.42·14-s + 0.625·15-s + 3.25·16-s − 1.25·17-s + 0.634·18-s − 0.755·19-s − 2.84·20-s + 1.03·21-s + 2.09·22-s − 1.88·23-s − 1.78·24-s + 0.175·25-s − 1.92·26-s − 0.192·27-s − 4.71·28-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 | \( 1 - 10.7T + 32T^{2} \) |
| 5 | \( 1 + 60.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 233.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 441.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 615.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.09e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.66e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.66e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.19e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.99e3T + 4.18e8T^{2} \) |
| 61 | \( 1 + 4.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.79e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.47e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.84e4T + 8.58e9T^{2} \) |
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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
−11.83256171692561757663039781064, −10.84470856492313200990950589655, −9.584901010732442112577685203886, −7.53530875317867733667913911507, −6.54534547003960793180106354875, −6.04049824818090472867320543782, −4.30191080897895582535347416008, −3.90668952206246203434348161009, −2.48553402020588528185654317513, 0,
2.48553402020588528185654317513, 3.90668952206246203434348161009, 4.30191080897895582535347416008, 6.04049824818090472867320543782, 6.54534547003960793180106354875, 7.53530875317867733667913911507, 9.584901010732442112577685203886, 10.84470856492313200990950589655, 11.83256171692561757663039781064
