| L(s) = 1 | − 1.52·2-s − 0.420·3-s + 0.325·4-s − 3.40·5-s + 0.641·6-s − 3.60·7-s + 2.55·8-s − 2.82·9-s + 5.18·10-s + 11-s − 0.137·12-s + 5.01·13-s + 5.49·14-s + 1.43·15-s − 4.54·16-s + 2.00·17-s + 4.30·18-s + 7.81·19-s − 1.10·20-s + 1.51·21-s − 1.52·22-s + 0.475·23-s − 1.07·24-s + 6.56·25-s − 7.64·26-s + 2.45·27-s − 1.17·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.242·3-s + 0.162·4-s − 1.52·5-s + 0.262·6-s − 1.36·7-s + 0.902·8-s − 0.940·9-s + 1.64·10-s + 0.301·11-s − 0.0395·12-s + 1.38·13-s + 1.46·14-s + 0.369·15-s − 1.13·16-s + 0.486·17-s + 1.01·18-s + 1.79·19-s − 0.247·20-s + 0.330·21-s − 0.325·22-s + 0.0992·23-s − 0.219·24-s + 1.31·25-s − 1.49·26-s + 0.471·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 3 | \( 1 + 0.420T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 - 0.475T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 3.79T + 43T^{2} \) |
| 47 | \( 1 + 7.04T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 0.581T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.242T + 83T^{2} \) |
| 89 | \( 1 + 8.37T + 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−9.228946746057953361771289292877, −8.272761940463627907663241418394, −7.81073057875114600896725120744, −6.90293508514743051553153547481, −6.05806148954208918340242062409, −4.90586655417130199100657003944, −3.58685169726946000724966430290, −3.26701776330137851392124265130, −1.05117387762641743796660994833, 0,
1.05117387762641743796660994833, 3.26701776330137851392124265130, 3.58685169726946000724966430290, 4.90586655417130199100657003944, 6.05806148954208918340242062409, 6.90293508514743051553153547481, 7.81073057875114600896725120744, 8.272761940463627907663241418394, 9.228946746057953361771289292877
