| L(s) = 1 | + (1.57 + 0.512i)2-s + (−2.99 − 0.181i)3-s + (−1.01 − 0.735i)4-s + (0.664 − 0.215i)5-s + (−4.62 − 1.81i)6-s + (2.11 + 1.53i)7-s + (−5.11 − 7.04i)8-s + (8.93 + 1.08i)9-s + 1.15·10-s + (2.89 + 2.38i)12-s + (−5.28 + 16.2i)13-s + (2.54 + 3.50i)14-s + (−2.02 + 0.526i)15-s + (−2.91 − 8.97i)16-s + (−15.3 + 4.97i)17-s + (13.5 + 6.28i)18-s + ⋯ |
| L(s) = 1 | + (0.788 + 0.256i)2-s + (−0.998 − 0.0603i)3-s + (−0.252 − 0.183i)4-s + (0.132 − 0.0431i)5-s + (−0.771 − 0.303i)6-s + (0.301 + 0.219i)7-s + (−0.639 − 0.880i)8-s + (0.992 + 0.120i)9-s + 0.115·10-s + (0.241 + 0.198i)12-s + (−0.406 + 1.25i)13-s + (0.181 + 0.250i)14-s + (−0.135 + 0.0350i)15-s + (−0.182 − 0.560i)16-s + (−0.901 + 0.292i)17-s + (0.751 + 0.349i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.237411 + 0.615531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.237411 + 0.615531i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.99 + 0.181i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.57 - 0.512i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (-0.664 + 0.215i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-2.11 - 1.53i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (5.28 - 16.2i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (15.3 - 4.97i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (12.8 - 9.36i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 - 23.1iT - 529T^{2} \) |
| 29 | \( 1 + (-3.15 + 4.34i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (1.25 - 3.86i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (51.3 + 37.3i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-39.6 - 54.5i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 22.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-43.3 - 59.7i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (40.7 + 13.2i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-16.8 + 23.2i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-14.3 - 44.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + 77.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-39.2 + 12.7i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (45.9 + 33.3i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-15.7 + 48.4i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-56.3 + 18.2i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + 38.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-5.03 + 15.4i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.70108267048301125654460433909, −10.85700340730219315914690008705, −9.738861896132620178076130063334, −9.000819431521724266286282319321, −7.43271284514667431148430608669, −6.42640238657293395159486088223, −5.71476868338580453159911241333, −4.71143714620283710708380253047, −3.96629429127240478744212787780, −1.74851663234610380880341802017,
0.25685624416305938509149528046, 2.46708706201758019800616779911, 4.02709956356687062305652149536, 4.84933818136691650660472519236, 5.68614615266171539259913024259, 6.77049223341903832709311352563, 7.983121720900294760618690049079, 9.054516485322022739428065886363, 10.35871392747439368788848578603, 10.92835393459418457377135241511
