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 − 1.44·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s − 3·11-s − 0.999·12-s + (−2.44 − 4.24i)13-s + (0.724 − 1.25i)14-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.775 − 1.34i)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 − 0.547·7-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 + (−0.679 − 1.17i)13-s + (0.193 − 0.335i)14-s + (0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.188 − 0.325i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.176425 - 0.335625i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.176425 - 0.335625i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.17 + 1.25i)T \) |
good | 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.775 + 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.94 - 5.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.67 + 8.09i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + (2.17 - 3.76i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.89 + 10.2i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.550 - 0.953i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.72 - 6.45i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.22 + 10.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.67 - 6.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.67 + 2.89i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.22 - 9.04i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.44 - 5.97i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + (-2.72 - 4.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.22 + 15.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.27513628734788854988141662902, −9.523076555515430079391525707826, −8.460283118625988891766630137652, −7.62440058200120831370906751720, −7.10580364137254428817306725166, −5.97681808692294597622231134292, −5.10916755571389116293082246517, −3.50442520325816704486954840895, −2.36383698899111400902639438441, −0.21731006135595366352541085317,
2.00783485135530071835925684591, 3.22182525093087833679656898344, 4.30379635883708533493032749221, 5.19595966259254896273159400494, 6.67663650933266709147450144050, 7.69473949047785222056410605547, 8.772184325426869532661805325246, 9.189139931182407019854089138941, 10.31396129336690662046538138710, 10.72925604717872236559527821837