| L(s) = 1 | + 3.34i·5-s + 4.73i·7-s + 4.24·11-s + 13-s − 3.34i·17-s + 1.26i·19-s + 7.34·23-s − 6.19·25-s + 4.00i·29-s + 6i·31-s − 15.8·35-s − 9.19·37-s − 7.34i·41-s − 2.19i·43-s − 3.10·47-s + ⋯ |
| L(s) = 1 | + 1.49i·5-s + 1.78i·7-s + 1.27·11-s + 0.277·13-s − 0.811i·17-s + 0.290i·19-s + 1.53·23-s − 1.23·25-s + 0.743i·29-s + 1.07i·31-s − 2.67·35-s − 1.51·37-s − 1.14i·41-s − 0.334i·43-s − 0.453·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724285953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724285953\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.34iT - 5T^{2} \) |
| 7 | \( 1 - 4.73iT - 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 3.34iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 - 4.00iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 + 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 2.19iT - 43T^{2} \) |
| 47 | \( 1 + 3.10T + 47T^{2} \) |
| 53 | \( 1 + 7.34iT - 53T^{2} \) |
| 59 | \( 1 - 3.10T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 - 7.26iT - 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 1.26iT - 79T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 + 9.38iT - 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.903207420447118860929405945855, −8.918945693187183846133208654517, −8.668223060798621266131281710450, −7.08495075827342139485539973847, −6.81185596757718268108869155096, −5.83202425095343767614438569375, −5.04747199821740287281771295743, −3.48995617241475416343943260040, −2.91098131368860403564471886859, −1.80848605431782376790338662490,
0.798419385698732741281378316009, 1.49084228384264020194594214846, 3.54428132894404643631857636815, 4.26774644918057330160718759934, 4.87010934043409725029943313785, 6.12797222324654949434677428128, 6.95297830496490686301523496758, 7.81458334584617568439465529932, 8.652366500884336926261294137508, 9.309568509981741064930333520149
