L(s) = 1 | + (−2.37 − 0.636i)2-s + (3.50 + 2.02i)4-s + (−0.498 − 0.498i)5-s + (−2.12 + 1.57i)7-s + (−3.56 − 3.56i)8-s + (0.866 + 1.50i)10-s + (−0.184 + 0.688i)11-s + (3.17 − 1.70i)13-s + (6.05 − 2.39i)14-s + (2.15 + 3.73i)16-s + (−2.27 + 3.93i)17-s + (−3.24 + 0.870i)19-s + (−0.739 − 2.75i)20-s + (0.876 − 1.51i)22-s + (1.67 − 0.964i)23-s + ⋯ |
L(s) = 1 | + (−1.68 − 0.450i)2-s + (1.75 + 1.01i)4-s + (−0.222 − 0.222i)5-s + (−0.802 + 0.596i)7-s + (−1.26 − 1.26i)8-s + (0.274 + 0.474i)10-s + (−0.0556 + 0.207i)11-s + (0.881 − 0.471i)13-s + (1.61 − 0.640i)14-s + (0.538 + 0.932i)16-s + (−0.551 + 0.954i)17-s + (−0.745 + 0.199i)19-s + (−0.165 − 0.616i)20-s + (0.186 − 0.323i)22-s + (0.348 − 0.201i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0652372 - 0.218279i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0652372 - 0.218279i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.12 - 1.57i)T \) |
| 13 | \( 1 + (-3.17 + 1.70i)T \) |
good | 2 | \( 1 + (2.37 + 0.636i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (0.498 + 0.498i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.184 - 0.688i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (2.27 - 3.93i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.24 - 0.870i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.67 + 0.964i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.185 + 0.322i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.53 + 3.53i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.545 - 2.03i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.11 + 11.6i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.38 - 3.68i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.55 - 3.55i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 + (1.03 + 3.85i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (10.0 + 5.81i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.5 + 3.37i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.10 + 7.86i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.608 - 0.608i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 + (2.25 + 2.25i)T + 83iT^{2} \) |
| 89 | \( 1 + (17.5 + 4.70i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (8.73 - 2.34i)T + (84.0 - 48.5i)T^{2} \) |
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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
−9.799788721055298983015279525102, −8.972450085174724366403232597713, −8.501864452631102802039176634858, −7.69679807553551543860560531248, −6.60331847070170604820358182352, −5.86982863032994885497661757137, −4.12814693328483957514976169332, −2.90118883815720111579620046999, −1.81139189408271824881007819063, −0.21098740703026117508648098305,
1.25151747859611814997457541814, 2.82762114977041367854082441475, 4.14039701109598925283276592589, 5.77012685933954324151299120629, 6.76666951932924250093914022684, 7.12634370938047645373109927961, 8.104810082109747221644762308864, 9.103441641964441671346220799652, 9.372374591196503712426124876058, 10.55528931543157325844819592961