L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (−0.900 + 0.433i)8-s + (0.826 − 0.563i)10-s + (0.365 − 0.930i)11-s + (0.365 − 0.930i)13-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s + (−0.5 − 0.866i)19-s + (0.955 + 0.294i)20-s + (0.955 − 0.294i)22-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.955 − 0.294i)26-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (−0.900 + 0.433i)8-s + (0.826 − 0.563i)10-s + (0.365 − 0.930i)11-s + (0.365 − 0.930i)13-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s + (−0.5 − 0.866i)19-s + (0.955 + 0.294i)20-s + (0.955 − 0.294i)22-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.955 − 0.294i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.763050726 + 0.01255982064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.763050726 + 0.01255982064i\) |
\(L(1)\) |
\(\approx\) |
\(1.402127059 + 0.2337968930i\) |
\(L(1)\) |
\(\approx\) |
\(1.402127059 + 0.2337968930i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 11 | \( 1 + (0.365 - 0.930i)T \) |
| 13 | \( 1 + (0.365 - 0.930i)T \) |
| 17 | \( 1 + (0.955 + 0.294i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.733 - 0.680i)T \) |
| 29 | \( 1 + (0.955 + 0.294i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.733 + 0.680i)T \) |
| 41 | \( 1 + (0.826 + 0.563i)T \) |
| 43 | \( 1 + (0.826 - 0.563i)T \) |
| 47 | \( 1 + (0.623 + 0.781i)T \) |
| 53 | \( 1 + (-0.733 - 0.680i)T \) |
| 59 | \( 1 + (-0.900 - 0.433i)T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.365 + 0.930i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.365 + 0.930i)T \) |
| 89 | \( 1 + (-0.988 - 0.149i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.628041676406981732861059580866, −23.035119589065479440354198213311, −22.4400325690685418600535951874, −21.34698221123086510724161343594, −20.94833298002359875710514322292, −19.67118646166382240955226456397, −19.074372433369839032792984827375, −18.27133798842441265407666363104, −17.39320409076682876894733844521, −15.97720623746494788983026124223, −15.06689302354470371472517373125, −14.17594908737485451004771464801, −13.80792968691292152749333137088, −12.298032383196349926069506264758, −11.88831017573715533897749712674, −10.75016912172837341235651659030, −10.03890333005711469056943552940, −9.2260300372887915536216069633, −7.66634867500010107773565854469, −6.56312372125647132661760099672, −5.801557282449649313529968889517, −4.408467299615358422468656249873, −3.64126941478483919998373169605, −2.47387599130317080414845539216, −1.53267662820050659911587739028,
0.89059309411630179795209905297, 2.78057358490670704618918650040, 3.88352246882320435752330082068, 4.87750462611653183498224403220, 5.77504579271788527729017151917, 6.55021482230007848301982085327, 8.10256714622347595529377541135, 8.36859673142948311341274091750, 9.51610414140041925369516074490, 10.882433902392213112769386611242, 12.11856257033468971290161555951, 12.68990629341580197927209981005, 13.65113669096387161255277297702, 14.31777810953604733775509678498, 15.58702021048039918824603763338, 16.07179958429122706624367634513, 17.0691965211102888224458448870, 17.57595312840318394388998062789, 18.84702510316327646688759291791, 19.96121076267181356107701426612, 20.88060909703072462611968722860, 21.54338863372900943136463773605, 22.43224939090173016302279086649, 23.41599919353981526915848761128, 24.08545529561448210590195739220