L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.900 − 0.433i)10-s + (−0.781 − 0.623i)11-s + (−0.781 − 0.623i)13-s + (−0.900 + 0.433i)16-s + (0.974 + 0.222i)17-s + i·19-s + (0.222 − 0.974i)20-s + (−0.974 + 0.222i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.974 + 0.222i)26-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.900 − 0.433i)10-s + (−0.781 − 0.623i)11-s + (−0.781 − 0.623i)13-s + (−0.900 + 0.433i)16-s + (0.974 + 0.222i)17-s + i·19-s + (0.222 − 0.974i)20-s + (−0.974 + 0.222i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.974 + 0.222i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.720799804 + 0.4372445262i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.720799804 + 0.4372445262i\) |
\(L(1)\) |
\(\approx\) |
\(1.306610012 - 0.4369553950i\) |
\(L(1)\) |
\(\approx\) |
\(1.306610012 - 0.4369553950i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.900 + 0.433i)T \) |
| 11 | \( 1 + (-0.781 - 0.623i)T \) |
| 13 | \( 1 + (-0.781 - 0.623i)T \) |
| 17 | \( 1 + (0.974 + 0.222i)T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.222 + 0.974i)T \) |
| 29 | \( 1 + (0.974 + 0.222i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 43 | \( 1 + (0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.781 - 0.623i)T \) |
| 53 | \( 1 + (-0.974 + 0.222i)T \) |
| 59 | \( 1 + (-0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.222 + 0.974i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.974 + 0.222i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.781 + 0.623i)T \) |
| 97 | \( 1 + iT \) |
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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
−17.48307804987471607569445582795, −17.0156488558097118903848086904, −16.15233449186872856357361617119, −15.93279635605537852419291807474, −14.80301279701115880209048174191, −14.4474539835393927894460390684, −13.8419196407744905359383938874, −13.05354532735305702281796819852, −12.57455260656643043505930913513, −12.12230316721149650919879677303, −11.0914839083588245414872281818, −10.2715306802765860612000424394, −9.419290476190380477769215020378, −9.0931071750495420207864702608, −8.1058233993618398939702839915, −7.48641991117477456189452943766, −6.7995314591750870756400985730, −6.136100164321883293762656253870, −5.33567903913774167733676196075, −4.79637678866900469606004436724, −4.37358633769943786595725075918, −3.07602877398450758062615458385, −2.5451325514380650895101172305, −1.731542235194873911338308453879, −0.335124710053468834559933676845,
1.10156995334638886342880936541, 1.72901444904850093760675743043, 2.72709808834176633505282181489, 3.085740406421463111992262273866, 3.84988888562957497608017477481, 5.02044650682108377399730480819, 5.47393858591203948192601884048, 5.92443283518465268199593201546, 6.81548301288310660718963867526, 7.70200247612439205997983163281, 8.46381639660971768874843483388, 9.58443348355044962657316313347, 9.821196603930596247367151634256, 10.693366096412204154302116595974, 10.867117276114693426718705768575, 12.0844192208805133819062599074, 12.396615091274642487725309275309, 13.275512608998447250168519680523, 13.69642543332618601993551417855, 14.401551857584684458808539727899, 14.87611593539066798782431410414, 15.59573041465088474671810781265, 16.46959491605346763171406792037, 17.21103040433985152623788586362, 17.96655250338374392374104326282