L(s) = 1 | + (−0.989 + 0.142i)3-s + (−0.755 − 0.654i)7-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.755 + 0.654i)13-s + (0.909 + 0.415i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (−0.909 + 0.415i)27-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (−0.755 + 0.654i)33-s + (0.281 + 0.959i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.142i)3-s + (−0.755 − 0.654i)7-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.755 + 0.654i)13-s + (0.909 + 0.415i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (−0.909 + 0.415i)27-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (−0.755 + 0.654i)33-s + (0.281 + 0.959i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0350 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0350 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5199135233 - 0.5020087539i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5199135233 - 0.5020087539i\) |
\(L(1)\) |
\(\approx\) |
\(0.7055356401 - 0.1133330635i\) |
\(L(1)\) |
\(\approx\) |
\(0.7055356401 - 0.1133330635i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 7 | \( 1 + (-0.755 - 0.654i)T \) |
| 11 | \( 1 + (0.841 - 0.540i)T \) |
| 13 | \( 1 + (-0.755 + 0.654i)T \) |
| 17 | \( 1 + (0.909 + 0.415i)T \) |
| 19 | \( 1 + (-0.415 - 0.909i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (-0.142 + 0.989i)T \) |
| 37 | \( 1 + (0.281 + 0.959i)T \) |
| 41 | \( 1 + (-0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.989 - 0.142i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.755 - 0.654i)T \) |
| 59 | \( 1 + (-0.654 - 0.755i)T \) |
| 61 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.540 - 0.841i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (-0.909 + 0.415i)T \) |
| 79 | \( 1 + (-0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.281 - 0.959i)T \) |
| 89 | \( 1 + (-0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.281 - 0.959i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.2320836587890597768194032204, −21.55330360343297969074735822558, −20.51866263752120023692010838164, −19.54921306966346492551182722075, −18.83406717411265874894833922323, −18.13509626841825881233674291921, −17.21962865908609378718628600170, −16.64718510170648182278827474552, −15.85581260826558224711883340807, −14.98696431744899782796885973864, −14.16784456461050867361325800871, −12.782459182238550060650604516524, −12.44654245590898614535595724680, −11.802364242089774546650132386088, −10.72514225858632463838546943603, −9.853582078403887616572269217347, −9.31696856973783493548078645118, −7.917321565296216073555100536803, −7.10142614953608441310390667657, −6.18251199967245524390487848347, −5.56309566423566808992696586980, −4.591967058617283082236603998883, −3.50717583951307755974696514912, −2.31078807062907078952526936872, −1.097042640791551025138778715897,
0.429483327464394120346868877787, 1.58201362539647097119859798281, 3.14844001615865160031047803605, 4.086572558609547899134254028267, 4.879063644639848271035492931146, 6.0287383918557848090293656791, 6.67137077900473748471534686156, 7.357294001026072387939982968116, 8.70992657838762194583656476852, 9.72102827308979448765332447009, 10.2546408191770744919885654871, 11.240220610021186813713557780694, 11.95315253231844833586305567472, 12.699465875114388027312709692946, 13.62416747191441659758369487327, 14.46924100915424787966163888488, 15.529037305278662922834156224562, 16.312651430081778844017331804266, 17.130380574855521795293422726521, 17.24470666836452282144254384952, 18.65333343118256369607987516224, 19.26011673398867817196423045791, 19.96144445094698827040385839005, 21.177457762831528282120661867022, 21.820821910714439476988464430275