L(s) = 1 | + (0.866 − 0.5i)2-s + (−1.68 − 0.416i)3-s + (0.499 − 0.866i)4-s + (0.798 + 1.38i)5-s + (−1.66 + 0.479i)6-s + (0.157 + 2.64i)7-s − 0.999i·8-s + (2.65 + 1.40i)9-s + (1.38 + 0.798i)10-s + (−0.866 − 0.5i)11-s + (−1.20 + 1.24i)12-s + 4.50i·13-s + (1.45 + 2.20i)14-s + (−0.765 − 2.65i)15-s + (−0.5 − 0.866i)16-s + (−3.12 + 5.40i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.970 − 0.240i)3-s + (0.249 − 0.433i)4-s + (0.356 + 0.618i)5-s + (−0.679 + 0.195i)6-s + (0.0593 + 0.998i)7-s − 0.353i·8-s + (0.884 + 0.466i)9-s + (0.437 + 0.252i)10-s + (−0.261 − 0.150i)11-s + (−0.346 + 0.360i)12-s + 1.25i·13-s + (0.389 + 0.590i)14-s + (−0.197 − 0.686i)15-s + (−0.125 − 0.216i)16-s + (−0.756 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47755 + 0.351262i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47755 + 0.351262i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (1.68 + 0.416i)T \) |
| 7 | \( 1 + (-0.157 - 2.64i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.798 - 1.38i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 4.50iT - 13T^{2} \) |
| 17 | \( 1 + (3.12 - 5.40i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.07 + 3.50i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.03 + 2.33i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.53iT - 29T^{2} \) |
| 31 | \( 1 + (-7.88 - 4.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.285 + 0.494i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.59T + 41T^{2} \) |
| 43 | \( 1 + 1.54T + 43T^{2} \) |
| 47 | \( 1 + (3.56 + 6.18i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.14 + 4.70i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.62 - 4.54i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.88 + 1.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.613 + 1.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.47iT - 71T^{2} \) |
| 73 | \( 1 + (-3.56 - 2.06i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.87 - 6.70i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + (-6.69 - 11.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.11iT - 97T^{2} \) |
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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
−11.21930821349121804488174696505, −10.58343296545118500880288760245, −9.567376200520251225037876998814, −8.484435231527606585661361033508, −6.81766673562995255685683551033, −6.50533796464958099942257941644, −5.36754538187488688631754473292, −4.58980952283299815601760103665, −2.95437798455944500814977950645, −1.72317194341736921576048991713,
0.951577705562741792895519416639, 3.21073233019680407678907262300, 4.58550660718251210810633705102, 5.15969012101080422318414239656, 6.08489225739027502099823862634, 7.21294073934208852026284586344, 7.892796747227496991409926770002, 9.474321594918326621918424614420, 10.10768709764023811067371067911, 11.18215526779078043218924649042