| L(s) = 1 | + 5.65i·5-s + 3·7-s + 5.65i·11-s − 17·13-s + 28.2i·17-s + 11·19-s + 39.5i·23-s − 7.00·25-s − 33.9i·29-s + 50·31-s + 16.9i·35-s − 33·37-s + 33.9i·41-s + 10·43-s − 84.8i·47-s + ⋯ |
| L(s) = 1 | + 1.13i·5-s + 0.428·7-s + 0.514i·11-s − 1.30·13-s + 1.66i·17-s + 0.578·19-s + 1.72i·23-s − 0.280·25-s − 1.17i·29-s + 1.61·31-s + 0.484i·35-s − 0.891·37-s + 0.827i·41-s + 0.232·43-s − 1.80i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.973328 + 0.973328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.973328 + 0.973328i\) |
| \(L(2)\) |
|
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 | \( 1 - 5.65iT - 25T^{2} \) |
| 7 | \( 1 - 3T + 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 17T + 169T^{2} \) |
| 17 | \( 1 - 28.2iT - 289T^{2} \) |
| 19 | \( 1 - 11T + 361T^{2} \) |
| 23 | \( 1 - 39.5iT - 529T^{2} \) |
| 29 | \( 1 + 33.9iT - 841T^{2} \) |
| 31 | \( 1 - 50T + 961T^{2} \) |
| 37 | \( 1 + 33T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 28.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 41T + 3.72e3T^{2} \) |
| 67 | \( 1 - 83T + 4.48e3T^{2} \) |
| 71 | \( 1 + 22.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 127T + 5.32e3T^{2} \) |
| 79 | \( 1 - 19T + 6.24e3T^{2} \) |
| 83 | \( 1 + 124. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 84.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 167T + 9.40e3T^{2} \) |
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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
−12.14726947026246155504215770669, −11.40599647432225203179388334938, −10.29223379469338757686925298953, −9.713383141607929016613142171665, −8.132605755575295443649256126213, −7.31507933911520968485166665514, −6.28003321720746245590593858284, −4.93509616168324422679468120437, −3.48488220339928317122219399230, −2.05405288164997694335014978967,
0.78174922194648870619651302241, 2.72031815038004948737886606450, 4.66844462646731653008030244918, 5.17754406266611476507523462822, 6.81170480450691788822144344305, 7.958718841447140589960764660766, 8.897651941695175288961062409735, 9.734215851575216606019678866892, 10.95509757886721116734760142887, 12.10814744032847541178956038954
