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
−10.72925604717872236559527821837, −10.31396129336690662046538138710, −9.189139931182407019854089138941, −8.772184325426869532661805325246, −7.69473949047785222056410605547, −6.67663650933266709147450144050, −5.19595966259254896273159400494, −4.30379635883708533493032749221, −3.22182525093087833679656898344, −2.00783485135530071835925684591,
0.21731006135595366352541085317, 2.36383698899111400902639438441, 3.50442520325816704486954840895, 5.10916755571389116293082246517, 5.97681808692294597622231134292, 7.10580364137254428817306725166, 7.62440058200120831370906751720, 8.460283118625988891766630137652, 9.523076555515430079391525707826, 10.27513628734788854988141662902