| L(s) = 1 | − 2.50i·2-s − 2.26·4-s − 1.29·5-s + 7.07i·7-s − 4.33i·8-s + 3.24i·10-s + 4.48i·13-s + 17.7·14-s − 19.9·16-s + 9.83i·17-s − 16.0i·19-s + 2.93·20-s + 39.6·23-s − 23.3·25-s + 11.2·26-s + ⋯ |
| L(s) = 1 | − 1.25i·2-s − 0.566·4-s − 0.259·5-s + 1.01i·7-s − 0.542i·8-s + 0.324i·10-s + 0.344i·13-s + 1.26·14-s − 1.24·16-s + 0.578i·17-s − 0.842i·19-s + 0.146·20-s + 1.72·23-s − 0.932·25-s + 0.431·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.781991647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781991647\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.50iT - 4T^{2} \) |
| 5 | \( 1 + 1.29T + 25T^{2} \) |
| 7 | \( 1 - 7.07iT - 49T^{2} \) |
| 13 | \( 1 - 4.48iT - 169T^{2} \) |
| 17 | \( 1 - 9.83iT - 289T^{2} \) |
| 19 | \( 1 + 16.0iT - 361T^{2} \) |
| 23 | \( 1 - 39.6T + 529T^{2} \) |
| 29 | \( 1 + 46.7iT - 841T^{2} \) |
| 31 | \( 1 - 34T + 961T^{2} \) |
| 37 | \( 1 - 11.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 65.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 78.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 66.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 97.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 39.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 43.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 75.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 61.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 61.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 103.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 10.6T + 9.40e3T^{2} \) |
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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
−9.399045449113213178514315464929, −8.973850352998559530818037487043, −7.907449988248885601848654634896, −6.83481889252738414875486142126, −5.95061371062316546588102942865, −4.78196180010230158957136223668, −3.85880050280369549369794000008, −2.77603318109329572761885317186, −2.07536001220083327632430261487, −0.65714633812204033393668032481,
1.04656216686159284258334886940, 2.81038091725714341502519372286, 4.03398827980070416944039572061, 4.98965944160312910685135573390, 5.79404429524163341839172715025, 6.89336341553255358123862173285, 7.25898787955261938442077015528, 8.080813361733064377018162350145, 8.819508429848937089115068805568, 9.854700052600545838788263887508
