| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 0.381·5-s + 3·7-s + 2.23·8-s − 0.618·10-s + 6.23·13-s − 4.85·14-s − 4.85·16-s + 0.618·17-s − 0.854·19-s + 0.236·20-s + 5.47·23-s − 4.85·25-s − 10.0·26-s + 1.85·28-s + 4.47·29-s − 3.85·31-s + 3.38·32-s − 1.00·34-s + 1.14·35-s − 4.23·37-s + 1.38·38-s + 0.854·40-s − 5.94·41-s + 1.76·43-s − 8.85·46-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.170·5-s + 1.13·7-s + 0.790·8-s − 0.195·10-s + 1.72·13-s − 1.29·14-s − 1.21·16-s + 0.149·17-s − 0.195·19-s + 0.0527·20-s + 1.14·23-s − 0.970·25-s − 1.97·26-s + 0.350·28-s + 0.830·29-s − 0.692·31-s + 0.597·32-s − 0.171·34-s + 0.193·35-s − 0.696·37-s + 0.224·38-s + 0.135·40-s − 0.928·41-s + 0.268·43-s − 1.30·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.083617264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083617264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 - 7.38T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 9.47T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
| 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
−9.810594438027469134619056506416, −8.726879803860141242320017408954, −8.549773179661251261034003107670, −7.66069687244334961841222307311, −6.77549609985789858096673811816, −5.63447000050821234618991322262, −4.68821999317995111248257246208, −3.63330394351187429002130763804, −1.94040847984149739027423933471, −1.03899719477631895175166871998,
1.03899719477631895175166871998, 1.94040847984149739027423933471, 3.63330394351187429002130763804, 4.68821999317995111248257246208, 5.63447000050821234618991322262, 6.77549609985789858096673811816, 7.66069687244334961841222307311, 8.549773179661251261034003107670, 8.726879803860141242320017408954, 9.810594438027469134619056506416
