L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.499 − 0.866i)6-s − 7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s − 3·11-s + 0.999·12-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)14-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (3 − 5.19i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.204 − 0.353i)6-s − 0.377·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s − 0.904·11-s + 0.288·12-s + (−0.277 − 0.480i)13-s + (0.133 − 0.231i)14-s + (0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.435427 - 0.311837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.435427 - 0.311837i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 18T + 83T^{2} \) |
| 89 | \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.16638017014507483516433699708, −9.889088692994453425513456718583, −8.750039238270802997566335987055, −7.993120986924548534762000072924, −6.96557143561581609103695248426, −5.84590951473238843458065623410, −5.21298337763800090190969209635, −4.11896752997089085795338377348, −2.54302185568816031686730999018, −0.34306327853343236205424443068,
1.68812285603811603991036572624, 2.83036909704991749660056451745, 4.07941694382071114517495335772, 5.50777839834613963862777190539, 6.38873088312627976819022714305, 7.51857904000476600701679172902, 8.160784941207707021441747817290, 9.402657941259542503965876920269, 10.09003637158575800221049725533, 10.94873816678968340129075164655