| L(s) = 1 | + (−0.453 − 0.891i)2-s + (−0.156 + 0.987i)3-s + (−0.587 + 0.809i)4-s + (2.02 + 0.946i)5-s + (0.951 − 0.309i)6-s + (1.76 − 0.279i)7-s + (0.987 + 0.156i)8-s + (−0.951 − 0.309i)9-s + (−0.0764 − 2.23i)10-s + (−1.65 + 2.87i)11-s + (−0.707 − 0.707i)12-s + (−4.74 + 2.41i)13-s + (−1.05 − 1.44i)14-s + (−1.25 + 1.85i)15-s + (−0.309 − 0.951i)16-s + (5.10 + 2.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (−0.0903 + 0.570i)3-s + (−0.293 + 0.404i)4-s + (0.906 + 0.423i)5-s + (0.388 − 0.126i)6-s + (0.667 − 0.105i)7-s + (0.349 + 0.0553i)8-s + (−0.317 − 0.103i)9-s + (−0.0241 − 0.706i)10-s + (−0.499 + 0.866i)11-s + (−0.204 − 0.204i)12-s + (−1.31 + 0.670i)13-s + (−0.280 − 0.386i)14-s + (−0.323 + 0.478i)15-s + (−0.0772 − 0.237i)16-s + (1.23 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19532 + 0.313941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19532 + 0.313941i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 + 0.891i)T \) |
| 3 | \( 1 + (0.156 - 0.987i)T \) |
| 5 | \( 1 + (-2.02 - 0.946i)T \) |
| 11 | \( 1 + (1.65 - 2.87i)T \) |
| good | 7 | \( 1 + (-1.76 + 0.279i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (4.74 - 2.41i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-5.10 - 2.59i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-5.99 + 4.35i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.06 - 3.06i)T - 23iT^{2} \) |
| 29 | \( 1 + (-6.00 - 4.36i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.63 + 5.02i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.661 + 4.17i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.957 + 1.31i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.98 + 1.98i)T + 43iT^{2} \) |
| 47 | \( 1 + (10.9 + 1.73i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-3.30 - 6.47i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.189 + 0.261i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-13.2 + 4.29i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (6.17 + 6.17i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.30 + 4.02i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.0629 + 0.397i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.91 + 12.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.799 + 1.56i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 16.7iT - 89T^{2} \) |
| 97 | \( 1 + (6.47 - 3.30i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.59140883230134086273640716071, −10.51362640370476745680114625924, −9.851268603626666608429261927132, −9.363326384571904989818357378084, −7.921944812137964256871030941899, −7.03124964927652717807018612315, −5.41922164402219445728536357277, −4.66972144256289069979707907529, −3.07346938251881482534091982373, −1.84294975601590185163196679803,
1.11086422348030823077395268028, 2.79038829699519767648026739301, 5.11718169755615075621638301552, 5.47771563270089779163652203145, 6.67974120720844198023887911679, 7.990959790561154485447306618477, 8.237141065134759881228910201524, 9.820468780554146864950200175198, 10.13760843605794992647123546021, 11.68888121068208145903951734237
