L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.62 − 2.09i)7-s − 0.999·8-s + (−2.12 − 3.67i)11-s − 2.24·13-s + (−0.999 − 2.44i)14-s + (−0.5 + 0.866i)16-s + (1.12 − 1.94i)19-s − 4.24·22-s + (−0.621 + 1.07i)23-s + (2.5 + 4.33i)25-s + (−1.12 + 1.94i)26-s + (−2.62 − 0.358i)28-s − 4.24·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.612 − 0.790i)7-s − 0.353·8-s + (−0.639 − 1.10i)11-s − 0.621·13-s + (−0.267 − 0.654i)14-s + (−0.125 + 0.216i)16-s + (0.257 − 0.445i)19-s − 0.904·22-s + (−0.129 + 0.224i)23-s + (0.5 + 0.866i)25-s + (−0.219 + 0.380i)26-s + (−0.495 − 0.0677i)28-s − 0.787·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.459130106\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.459130106\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.62 + 2.09i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 + 1.94i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.621 - 1.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + (4.62 + 8.00i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + (-2.37 + 4.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.12 + 3.67i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 1.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.121 + 0.210i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.378 - 0.655i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + (-5.74 + 9.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−9.612992434971733077437116807127, −8.746762482947005048607639475440, −7.76694983138076038356190790646, −7.10245653205509598825588926733, −5.76014690662713556313166318217, −5.13527935686452325792742372550, −4.08575682002091788245698685917, −3.20195409153305872503677268307, −1.99768441089550463824730956659, −0.54855713644822122429342538289,
1.91615420627963810915629528665, 2.97972092985385816734172750078, 4.41106530349029757391005693410, 5.05717189232371343342749659264, 5.81760743950951191597269671899, 6.93073373174175181996964836609, 7.61839792398702174707337967269, 8.391690889602323213716627223058, 9.213074218870219735221597525291, 10.08535024931018577412225806640