| L(s) = 1 | + 4.82·2-s − 0.0989·3-s − 8.73·4-s − 25·5-s − 0.477·6-s − 108.·7-s − 196.·8-s − 242.·9-s − 120.·10-s + 121·11-s + 0.864·12-s + 1.11e3·13-s − 521.·14-s + 2.47·15-s − 667.·16-s + 599.·17-s − 1.17e3·18-s + 361·19-s + 218.·20-s + 10.7·21-s + 583.·22-s + 2.33e3·23-s + 19.4·24-s + 625·25-s + 5.39e3·26-s + 48.0·27-s + 945.·28-s + ⋯ |
| L(s) = 1 | + 0.852·2-s − 0.00634·3-s − 0.273·4-s − 0.447·5-s − 0.00541·6-s − 0.834·7-s − 1.08·8-s − 0.999·9-s − 0.381·10-s + 0.301·11-s + 0.00173·12-s + 1.83·13-s − 0.711·14-s + 0.00283·15-s − 0.652·16-s + 0.503·17-s − 0.852·18-s + 0.229·19-s + 0.122·20-s + 0.00529·21-s + 0.257·22-s + 0.922·23-s + 0.00688·24-s + 0.200·25-s + 1.56·26-s + 0.0126·27-s + 0.227·28-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 | \( 1 - 4.82T + 32T^{2} \) |
| 3 | \( 1 + 0.0989T + 243T^{2} \) |
| 7 | \( 1 + 108.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 599.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.33e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 576.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 126.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.47e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.58e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.41e5T + 8.58e9T^{2} \) |
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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
−8.856710227064230866027405168593, −8.085549145957638016172908790923, −6.78593094339618399994140421828, −6.01509191745850361529436834014, −5.42611438900514069176645994554, −4.23094357052755147962443472740, −3.43269123926575191971926471219, −2.93936917524403993018831903474, −1.07791609755484122156185219515, 0,
1.07791609755484122156185219515, 2.93936917524403993018831903474, 3.43269123926575191971926471219, 4.23094357052755147962443472740, 5.42611438900514069176645994554, 6.01509191745850361529436834014, 6.78593094339618399994140421828, 8.085549145957638016172908790923, 8.856710227064230866027405168593
