| L(s) = 1 | + (−0.750 + 1.47i)2-s + (−1.38 + 1.03i)3-s + (−0.432 − 0.594i)4-s + (−0.103 + 0.651i)5-s + (−0.484 − 2.82i)6-s + (−1.17 + 1.38i)7-s + (−2.06 + 0.327i)8-s + (0.853 − 2.87i)9-s + (−0.882 − 0.641i)10-s + (−0.889 − 1.45i)11-s + (1.21 + 0.377i)12-s + (−0.0823 + 1.04i)13-s + (−1.14 − 2.77i)14-s + (−0.531 − 1.01i)15-s + (1.52 − 4.68i)16-s + (−0.540 + 2.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.530 + 1.04i)2-s + (−0.801 + 0.598i)3-s + (−0.216 − 0.297i)4-s + (−0.0461 + 0.291i)5-s + (−0.197 − 1.15i)6-s + (−0.445 + 0.521i)7-s + (−0.730 + 0.115i)8-s + (0.284 − 0.958i)9-s + (−0.279 − 0.202i)10-s + (−0.268 − 0.437i)11-s + (0.350 + 0.109i)12-s + (−0.0228 + 0.290i)13-s + (−0.307 − 0.741i)14-s + (−0.137 − 0.261i)15-s + (0.380 − 1.17i)16-s + (−0.131 + 0.546i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0349989 - 0.507886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0349989 - 0.507886i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.38 - 1.03i)T \) |
| 41 | \( 1 + (-6.39 + 0.327i)T \) |
| good | 2 | \( 1 + (0.750 - 1.47i)T + (-1.17 - 1.61i)T^{2} \) |
| 5 | \( 1 + (0.103 - 0.651i)T + (-4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (1.17 - 1.38i)T + (-1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (0.889 + 1.45i)T + (-4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (0.0823 - 1.04i)T + (-12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (0.540 - 2.25i)T + (-15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.405 - 5.14i)T + (-18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-1.68 - 5.19i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.192 + 0.799i)T + (-25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (3.67 - 5.05i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.58 + 3.33i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (7.88 + 4.01i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (-5.81 + 4.96i)T + (7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (2.73 - 0.655i)T + (47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-12.0 + 3.91i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.93 + 11.6i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (6.07 - 9.91i)T + (-30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (8.13 - 4.98i)T + (32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-3.53 + 3.53i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.58 + 6.24i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 6.34iT - 83T^{2} \) |
| 89 | \( 1 + (-7.77 - 6.64i)T + (13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (-8.27 - 5.07i)T + (44.0 + 86.4i)T^{2} \) |
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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
−14.45625575443900988299266983905, −12.77229218631826659311858488642, −11.83446779967555326551735936184, −10.76016769261166328374033387108, −9.585823926664366557274902442314, −8.679381468817777984048587995586, −7.30859347466682480790730602897, −6.22026193009151795689446186704, −5.42977165436737070818472006981, −3.45804216818977011867373748402,
0.70828700861844783040920989674, 2.61044824218618390865276231693, 4.72955614397934803270150965354, 6.28052558676784246636815257437, 7.38253872987174975017208342115, 8.912744879665524353609960832865, 10.07673570259237886475772652508, 10.87659808677963046939991071497, 11.71204667321022570658812237759, 12.74194813508263112173210360383
