L(s) = 1 | + (2.72 + 2.72i)2-s + 10.8i·4-s + (−4.39 + 2.38i)5-s + (−1.87 − 1.87i)7-s + (−18.6 + 18.6i)8-s + (−18.4 − 5.47i)10-s − 3.42·11-s + (7.98 − 7.98i)13-s − 10.1i·14-s − 57.9·16-s + (16.5 + 16.5i)17-s + 1.38i·19-s + (−25.8 − 47.6i)20-s + (−9.31 − 9.31i)22-s + (−18.8 + 18.8i)23-s + ⋯ |
L(s) = 1 | + (1.36 + 1.36i)2-s + 2.70i·4-s + (−0.879 + 0.476i)5-s + (−0.267 − 0.267i)7-s + (−2.32 + 2.32i)8-s + (−1.84 − 0.547i)10-s − 0.311·11-s + (0.613 − 0.613i)13-s − 0.727i·14-s − 3.62·16-s + (0.974 + 0.974i)17-s + 0.0726i·19-s + (−1.29 − 2.38i)20-s + (−0.423 − 0.423i)22-s + (−0.819 + 0.819i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.262i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.317328 - 2.37947i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.317328 - 2.37947i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (4.39 - 2.38i)T \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
good | 2 | \( 1 + (-2.72 - 2.72i)T + 4iT^{2} \) |
| 11 | \( 1 + 3.42T + 121T^{2} \) |
| 13 | \( 1 + (-7.98 + 7.98i)T - 169iT^{2} \) |
| 17 | \( 1 + (-16.5 - 16.5i)T + 289iT^{2} \) |
| 19 | \( 1 - 1.38iT - 361T^{2} \) |
| 23 | \( 1 + (18.8 - 18.8i)T - 529iT^{2} \) |
| 29 | \( 1 - 45.7iT - 841T^{2} \) |
| 31 | \( 1 - 43.2T + 961T^{2} \) |
| 37 | \( 1 + (-2.18 - 2.18i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 6.61T + 1.68e3T^{2} \) |
| 43 | \( 1 + (44.1 - 44.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (14.2 + 14.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-44.4 + 44.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 17.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-40.9 - 40.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 38.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-66.4 + 66.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 5.30iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-62.5 + 62.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 44.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (30.5 + 30.5i)T + 9.40e3iT^{2} \) |
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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
−12.27976243764021303748450116820, −11.41438950824909692103087454050, −10.21027496824418958743705325114, −8.357016771533141535967447331148, −7.942177026145561905079474561138, −6.93563169753740529969170739498, −6.09429587812255161132275611017, −5.05230323442528428036800185053, −3.78500660253127400846729854383, −3.21534474310435591494681870055,
0.75570084263231759557624608616, 2.45806559891100151115797120252, 3.64660647054509673418940111912, 4.50444352813708410926373201372, 5.49323444577220401045204726572, 6.61601391009470322942695476414, 8.221922870130651440146365414688, 9.483570965590929581784651678952, 10.28339970199056495491655506854, 11.41518015895013114310795656284