L(s) = 1 | + (−0.5 + 0.866i)3-s − 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s + 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s − 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s + 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1063618709 - 0.1376971945i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1063618709 - 0.1376971945i\) |
\(L(1)\) |
\(\approx\) |
\(0.5940886554 + 0.1632753463i\) |
\(L(1)\) |
\(\approx\) |
\(0.5940886554 + 0.1632753463i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.76912272160480054923618405036, −28.977370797293421143899313146875, −27.65109166206156463111336236075, −26.9484818413248893807801138148, −25.64766246130870855281154674904, −24.165597006987811729156304123496, −23.74619755161834181109286779765, −22.926295810406278825815474299442, −21.5538633748810307627838534318, −20.12082394744342097100259275990, −19.27323525478836914581763216062, −18.313242120109969796569205169637, −17.09461333488352122949314833217, −16.24913698009209615206643463339, −14.78471399209066712643112877622, −13.54067185358479051721915731383, −12.540492483571958258220223288489, −11.2774446497121714128940766110, −10.680873510828472431905033538473, −8.39009143360655193906214765384, −7.67913108913820139348248905774, −6.481171287991067464872883548908, −4.968454736790499123644963744174, −3.47783636632375730785943321104, −1.41468158242385770488805718284,
0.08411986133245054720994378796, 2.68486877521757785473401917745, 4.3733852460211915777991201951, 5.13450707579130745836325801173, 6.82139315423648894533401586957, 8.32493325215845953722271105448, 9.42364912718815163711893867176, 10.88427184403274438530649093259, 11.66012942230346853490518546841, 12.708383683079704550171050906225, 14.73132970085012457595790689263, 15.37795712036816480169719158816, 16.21410169830345824792633488197, 17.585905294877949674859230826701, 18.53977636868290620552973125956, 19.99674204320850640721330241432, 20.88496386042209911465839462536, 21.99611932897363211946083713411, 22.93693818255658800994487110139, 23.79988537780303966265905454619, 25.09494944675192453864232564857, 26.43599956693794714972648232950, 27.25010001348416179040472765632, 28.16347437125625367836539858567, 28.74277962505070557904353662098