L(s) = 1 | + (−0.5 − 0.866i)5-s + (2 − 3.46i)11-s + (2.5 + 4.33i)13-s + 5·17-s − 8·19-s + (2 + 3.46i)23-s + (2 − 3.46i)25-s + (1.5 − 2.59i)29-s + (−2 − 3.46i)31-s + 3·37-s + (−3 − 5.19i)41-s + (2 − 3.46i)43-s + (6 − 10.3i)47-s + (3.5 + 6.06i)49-s + 10·53-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (0.603 − 1.04i)11-s + (0.693 + 1.20i)13-s + 1.21·17-s − 1.83·19-s + (0.417 + 0.722i)23-s + (0.400 − 0.692i)25-s + (0.278 − 0.482i)29-s + (−0.359 − 0.622i)31-s + 0.493·37-s + (−0.468 − 0.811i)41-s + (0.304 − 0.528i)43-s + (0.875 − 1.51i)47-s + (0.5 + 0.866i)49-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.650044584\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.650044584\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 5T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2 - 3.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.385475331281543738792444858814, −8.695241142651462159968613424870, −8.195654278467891283764353648090, −7.00981787239066710511048951489, −6.24762789557569416874692823116, −5.46219823697741371143258584228, −4.18302797584311273586798964579, −3.68312173394666194702710747581, −2.18617020939688200816576879836, −0.839928882393009952657266570744,
1.19263162880615176554122155046, 2.63047069247772097442160373466, 3.63042180284140369905882268326, 4.56188882745564146067866458173, 5.61187139397256587851989597413, 6.52532271974236785181179119631, 7.25580295658511994284113733098, 8.147332612990981336412139005847, 8.856811894839119609351179393046, 9.846950304092217956126204920928