| L(s) = 1 | − 2.09·2-s + 3-s + 2.39·4-s − 1.55·5-s − 2.09·6-s + 4.13·7-s − 0.819·8-s + 9-s + 3.26·10-s + 4.43·11-s + 2.39·12-s + 3.67·13-s − 8.66·14-s − 1.55·15-s − 3.06·16-s − 2.09·18-s + 0.406·19-s − 3.72·20-s + 4.13·21-s − 9.28·22-s − 6.47·23-s − 0.819·24-s − 2.57·25-s − 7.70·26-s + 27-s + 9.88·28-s − 4.48·29-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.19·4-s − 0.696·5-s − 0.855·6-s + 1.56·7-s − 0.289·8-s + 0.333·9-s + 1.03·10-s + 1.33·11-s + 0.690·12-s + 1.02·13-s − 2.31·14-s − 0.402·15-s − 0.766·16-s − 0.493·18-s + 0.0932·19-s − 0.833·20-s + 0.902·21-s − 1.97·22-s − 1.35·23-s − 0.167·24-s − 0.514·25-s − 1.51·26-s + 0.192·27-s + 1.86·28-s − 0.833·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080677636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080677636\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 19 | \( 1 - 0.406T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 0.713T + 37T^{2} \) |
| 41 | \( 1 + 2.00T + 41T^{2} \) |
| 43 | \( 1 + 4.13T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 6.11T + 59T^{2} \) |
| 61 | \( 1 + 5.33T + 61T^{2} \) |
| 67 | \( 1 - 6.53T + 67T^{2} \) |
| 71 | \( 1 + 1.63T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 9.33T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.978991803582612312831479484981, −9.112790368027276052166596904776, −8.319290902312090786563912183724, −8.084765425552740552611318939455, −7.22575025556739939574854709879, −6.15173392490360537851631643464, −4.52838497375151842276165263877, −3.79899161951646548034361210324, −2.00544244509842489437304003198, −1.13628127481106249316103554823,
1.13628127481106249316103554823, 2.00544244509842489437304003198, 3.79899161951646548034361210324, 4.52838497375151842276165263877, 6.15173392490360537851631643464, 7.22575025556739939574854709879, 8.084765425552740552611318939455, 8.319290902312090786563912183724, 9.112790368027276052166596904776, 9.978991803582612312831479484981
