| L(s) = 1 | + (1.10 + 0.868i)2-s + (−0.310 + 0.436i)3-s + (−0.00622 − 0.0256i)4-s + (3.76 − 0.725i)5-s + (−0.721 + 0.211i)6-s + (−2.57 − 0.626i)7-s + (1.18 − 2.58i)8-s + (0.887 + 2.56i)9-s + (4.79 + 2.46i)10-s + (−1.38 + 1.09i)11-s + (0.0131 + 0.00525i)12-s + (−4.88 + 3.14i)13-s + (−2.29 − 2.92i)14-s + (−0.853 + 1.86i)15-s + (3.50 − 1.80i)16-s + (−1.89 − 1.80i)17-s + ⋯ |
| L(s) = 1 | + (0.780 + 0.614i)2-s + (−0.179 + 0.251i)3-s + (−0.00311 − 0.0128i)4-s + (1.68 − 0.324i)5-s + (−0.294 + 0.0865i)6-s + (−0.971 − 0.236i)7-s + (0.418 − 0.915i)8-s + (0.295 + 0.854i)9-s + (1.51 + 0.780i)10-s + (−0.418 + 0.329i)11-s + (0.00379 + 0.00151i)12-s + (−1.35 + 0.871i)13-s + (−0.613 − 0.781i)14-s + (−0.220 + 0.482i)15-s + (0.876 − 0.452i)16-s + (−0.460 − 0.438i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64800 + 0.485833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64800 + 0.485833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.57 + 0.626i)T \) |
| 23 | \( 1 + (-2.40 + 4.14i)T \) |
| good | 2 | \( 1 + (-1.10 - 0.868i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.310 - 0.436i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-3.76 + 0.725i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (1.38 - 1.09i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (4.88 - 3.14i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.89 + 1.80i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.993 - 0.947i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-2.91 + 0.856i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (7.51 - 0.717i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (0.296 + 0.857i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-3.22 - 3.72i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.291 + 0.638i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (0.390 - 0.677i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0886 - 1.86i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-1.87 - 0.968i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (5.02 + 7.06i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-11.8 + 4.73i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.96 - 13.6i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.119 - 0.494i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.0770 + 1.61i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.916 + 1.05i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-18.5 - 1.76i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (2.28 + 2.63i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−13.09967771486192627988004937613, −12.61972512554485530256024152420, −10.62024547678091909297169224096, −9.883783401401891280350508958113, −9.293958956599269110874084681139, −7.20696767288842546323505093552, −6.38466338781521258080363381205, −5.27540433347590047979547778602, −4.56682551451213609651875289777, −2.26632306863756560987217011806,
2.28382253413190360619512483517, 3.32267492989975842562728314552, 5.21314540713667154469710665498, 6.06179485015551465695441558216, 7.23772980671456387924042507142, 9.039644497578000239435796997151, 9.916544796785757673659560139222, 10.79776123402476333712091822895, 12.20378336173620637489462305814, 12.94841501584287551853245860687
