L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 0.866i)6-s + (−1 + 1.73i)7-s + 0.999·8-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)11-s + 1.73i·12-s + (−1 − 1.73i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s − 3·17-s − 3·18-s − 19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.612 − 0.353i)6-s + (−0.377 + 0.654i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.452 − 0.783i)11-s + 0.499i·12-s + (−0.277 − 0.480i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s − 0.727·17-s − 0.707·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.410344 + 0.0723547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.410344 + 0.0723547i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−18.65330191638767616820202141858, −17.58588719176622164211034015698, −16.50555445570130060214767610978, −15.36554110674836471808657271021, −13.60256088276205419926735569181, −12.18420205922030737509979676105, −10.66744030764575546769971244074, −8.790108040571259251135981778049, −6.92583624002767656428145584125, −5.52147912858687978615646302053,
4.36179441556210697223984606647, 6.85650392855044452343093046473, 9.327762105443371388541282310587, 10.51435608625116348572044517930, 11.75914826730233688681995342362, 13.09153056920829555057476441677, 15.04331307163907501027393190909, 16.65829432309937211505679158926, 17.34259594350077650929836362354, 18.73425098025001598515848722623