L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (5.59 + 9.69i)5-s + (−9.19 + 15.9i)7-s − 7.99·8-s + 22.3·10-s + (−11.7 + 20.4i)11-s + (33.8 + 58.6i)13-s + (18.3 + 31.8i)14-s + (−8 + 13.8i)16-s − 117.·17-s + 110.·19-s + (22.3 − 38.7i)20-s + (23.5 + 40.8i)22-s + (34.6 + 59.9i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.500 + 0.867i)5-s + (−0.496 + 0.860i)7-s − 0.353·8-s + 0.708·10-s + (−0.323 + 0.559i)11-s + (0.722 + 1.25i)13-s + (0.351 + 0.608i)14-s + (−0.125 + 0.216i)16-s − 1.67·17-s + 1.33·19-s + (0.250 − 0.433i)20-s + (0.228 + 0.395i)22-s + (0.313 + 0.543i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.59374 + 0.743176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59374 + 0.743176i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-5.59 - 9.69i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (9.19 - 15.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (11.7 - 20.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-33.8 - 58.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-34.6 - 59.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-99.1 + 171. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-155. - 269. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (66.3 + 114. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-167. + 290. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (189. - 328. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (168. + 292. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (138. - 240. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (332. + 575. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 73.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-239. + 415. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (89.8 - 155. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 846.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-336. + 582. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.42526840391554126567091272279, −11.57229916240699096280881285866, −10.64314225060835278800021598196, −9.602727933263857029366308128047, −8.803316782762476577126355291837, −6.92262715704729926057755687537, −6.12623507871714322945051037723, −4.70011602710937871464307076658, −3.10525931143210135911954977396, −1.99834062961252519655337517313,
0.74933473418558299744814902238, 3.19432041675979517994047283245, 4.65601391033279153591822417531, 5.72606297497477389862627349745, 6.82597914966550234224095616781, 8.105387781285197801906498311118, 8.998368759341429306530200994417, 10.17538725441507789439260765338, 11.26401264430716138666016889340, 12.77110875044862782119880420778