| L(s) = 1 | − 1.49·2-s + 2.98·3-s + 0.230·4-s − 3.68·5-s − 4.45·6-s − 3.54·7-s + 2.64·8-s + 5.90·9-s + 5.49·10-s + 3.04·11-s + 0.687·12-s + 13-s + 5.29·14-s − 10.9·15-s − 4.40·16-s + 7.25·17-s − 8.82·18-s − 6.13·19-s − 0.848·20-s − 10.5·21-s − 4.55·22-s − 7.92·23-s + 7.88·24-s + 8.55·25-s − 1.49·26-s + 8.68·27-s − 0.816·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 1.72·3-s + 0.115·4-s − 1.64·5-s − 1.81·6-s − 1.33·7-s + 0.934·8-s + 1.96·9-s + 1.73·10-s + 0.919·11-s + 0.198·12-s + 0.277·13-s + 1.41·14-s − 2.83·15-s − 1.10·16-s + 1.75·17-s − 2.07·18-s − 1.40·19-s − 0.189·20-s − 2.30·21-s − 0.970·22-s − 1.65·23-s + 1.61·24-s + 1.71·25-s − 0.292·26-s + 1.67·27-s − 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 + 0.586T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 0.603T + 59T^{2} \) |
| 61 | \( 1 - 0.0966T + 61T^{2} \) |
| 67 | \( 1 - 4.60T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 9.23T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 + 6.20T + 89T^{2} \) |
| 97 | \( 1 + 3.97T + 97T^{2} \) |
| 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
−9.010884900467450032935105281498, −8.407160647159692717398862845204, −7.931129660808065523250517961014, −7.28818939818972991970757366589, −6.38238618169278001977459993558, −4.35477156677737274946925192397, −3.72418368527542166413162367477, −3.19797522563572377138014568308, −1.62163099709047228973640589646, 0,
1.62163099709047228973640589646, 3.19797522563572377138014568308, 3.72418368527542166413162367477, 4.35477156677737274946925192397, 6.38238618169278001977459993558, 7.28818939818972991970757366589, 7.931129660808065523250517961014, 8.407160647159692717398862845204, 9.010884900467450032935105281498
