| L(s) = 1 | + 2.79·2-s + 3-s + 5.78·4-s − 3.55·5-s + 2.79·6-s − 2.06·7-s + 10.5·8-s + 9-s − 9.93·10-s − 1.29·11-s + 5.78·12-s + 5.72·13-s − 5.75·14-s − 3.55·15-s + 17.9·16-s + 17-s + 2.79·18-s + 2.32·19-s − 20.5·20-s − 2.06·21-s − 3.60·22-s + 6.73·23-s + 10.5·24-s + 7.67·25-s + 15.9·26-s + 27-s − 11.9·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 0.577·3-s + 2.89·4-s − 1.59·5-s + 1.13·6-s − 0.780·7-s + 3.73·8-s + 0.333·9-s − 3.14·10-s − 0.389·11-s + 1.67·12-s + 1.58·13-s − 1.53·14-s − 0.919·15-s + 4.47·16-s + 0.242·17-s + 0.657·18-s + 0.533·19-s − 4.60·20-s − 0.450·21-s − 0.768·22-s + 1.40·23-s + 2.15·24-s + 1.53·25-s + 3.13·26-s + 0.192·27-s − 2.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.726016540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.726016540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 - 7.64T + 37T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 + 0.0340T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 - 0.858T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 2.96T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.262521761377305075173807960261, −7.84356432134845077974895282320, −6.92114724033802773088550793948, −6.51569628987782834719373444804, −5.45360804242215340205612158272, −4.64106313249571405980487406030, −3.83743671961590196581894555446, −3.32482444215078748551956814504, −2.87215209855287708181990059144, −1.27983655924897784773135360527,
1.27983655924897784773135360527, 2.87215209855287708181990059144, 3.32482444215078748551956814504, 3.83743671961590196581894555446, 4.64106313249571405980487406030, 5.45360804242215340205612158272, 6.51569628987782834719373444804, 6.92114724033802773088550793948, 7.84356432134845077974895282320, 8.262521761377305075173807960261
