| L(s) = 1 | + (−1.57 + 0.512i)2-s + (−0.753 − 2.90i)3-s + (−1.01 + 0.735i)4-s + (−0.664 − 0.215i)5-s + (2.67 + 4.19i)6-s + (2.11 − 1.53i)7-s + (5.11 − 7.04i)8-s + (−7.86 + 4.37i)9-s + 1.15·10-s + (2.89 + 2.38i)12-s + (−5.28 − 16.2i)13-s + (−2.54 + 3.50i)14-s + (−0.126 + 2.09i)15-s + (−2.91 + 8.97i)16-s + (15.3 + 4.97i)17-s + (10.1 − 10.9i)18-s + ⋯ |
| L(s) = 1 | + (−0.788 + 0.256i)2-s + (−0.251 − 0.967i)3-s + (−0.252 + 0.183i)4-s + (−0.132 − 0.0431i)5-s + (0.445 + 0.698i)6-s + (0.301 − 0.219i)7-s + (0.639 − 0.880i)8-s + (−0.873 + 0.486i)9-s + 0.115·10-s + (0.241 + 0.198i)12-s + (−0.406 − 1.25i)13-s + (−0.181 + 0.250i)14-s + (−0.00842 + 0.139i)15-s + (−0.182 + 0.560i)16-s + (0.901 + 0.292i)17-s + (0.564 − 0.607i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00106554 + 0.00320925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00106554 + 0.00320925i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.753 + 2.90i)T \) |
| 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
−10.52128382694473414679178211426, −9.706358967185312172934586074046, −8.399462112687125465736379914458, −7.928417912892315738095345739550, −7.19271623713919050093074344993, −6.00690338489860953355773289710, −4.80289843288727785609565605260, −3.21334843469633528465820356767, −1.37846956967135771874599908094, −0.00219611808759956971861782557,
1.98209486308606667053659842421, 3.80696230734191238236644744437, 4.84828347750469003617429884301, 5.73418694386596066215746858341, 7.21613719131188053215770438037, 8.552716168097894775801585451821, 9.014391732673807954372021826796, 10.04080132367206224888385736287, 10.52407237035120936170961586818, 11.56831136613803225877126404697
