| L(s) = 1 | + (1.90 − 0.619i)2-s + (−1.63 − 2.51i)3-s + (0.0135 − 0.00987i)4-s + (5.21 + 1.69i)5-s + (−4.67 − 3.78i)6-s + (−4.52 + 3.28i)7-s + (−4.69 + 6.45i)8-s + (−3.66 + 8.22i)9-s + 10.9·10-s + (2.23 − 10.7i)11-s + (−0.0470 − 0.0180i)12-s + (−3.00 − 9.24i)13-s + (−6.58 + 9.06i)14-s + (−4.25 − 15.8i)15-s + (−4.96 + 15.2i)16-s + (16.9 + 5.52i)17-s + ⋯ |
| L(s) = 1 | + (0.953 − 0.309i)2-s + (−0.544 − 0.838i)3-s + (0.00339 − 0.00246i)4-s + (1.04 + 0.338i)5-s + (−0.778 − 0.630i)6-s + (−0.646 + 0.469i)7-s + (−0.586 + 0.807i)8-s + (−0.406 + 0.913i)9-s + 1.09·10-s + (0.203 − 0.979i)11-s + (−0.00392 − 0.00150i)12-s + (−0.230 − 0.710i)13-s + (−0.470 + 0.647i)14-s + (−0.283 − 1.05i)15-s + (−0.310 + 0.955i)16-s + (0.999 + 0.324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.25316 - 0.399478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25316 - 0.399478i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.63 + 2.51i)T \) |
| 11 | \( 1 + (-2.23 + 10.7i)T \) |
| good | 2 | \( 1 + (-1.90 + 0.619i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (-5.21 - 1.69i)T + (20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (4.52 - 3.28i)T + (15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (3.00 + 9.24i)T + (-136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-16.9 - 5.52i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (15.0 + 10.9i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 + 12.3iT - 529T^{2} \) |
| 29 | \( 1 + (1.45 + 2.00i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-15.2 - 46.8i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-31.8 + 23.1i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (33.2 - 45.7i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + 43.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.9 + 46.7i)T + (-682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-41.0 + 13.3i)T + (2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-52.9 - 72.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (9.53 - 29.3i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 34.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (35.7 + 11.6i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-9.81 + 7.12i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (19.5 + 60.0i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (9.22 + 2.99i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 + 34.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (11.6 + 35.9i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−16.60387930948094253381705629391, −14.70465867790606122315194338825, −13.62179476106867535234581549027, −12.90471829486430214856393067692, −11.88943698816893532297853039156, −10.41419738038276192292633382193, −8.531682389078934187185456515212, −6.35378772470517259878756688678, −5.46674441313917918928535994250, −2.80683410649141799733119501045,
4.06239140290557108042790310609, 5.40887361338099353619873721221, 6.56955743871093617745612669867, 9.548048878464864925281611963152, 9.955695616487394603968570102368, 12.02081740028019086296148933310, 13.16408443771375591747199046727, 14.27412412202267055876902780684, 15.27092670358914085928211681759, 16.59960070120143384693711713047
