| L(s) = 1 | + (−1.94 − 0.866i)2-s + (1.86 − 1.35i)3-s + (1.69 + 1.88i)4-s + (−1.20 + 0.255i)5-s + (−4.81 + 1.02i)6-s + (0.404 − 3.84i)7-s + (−0.352 − 1.08i)8-s + (0.722 − 2.22i)9-s + (2.56 + 0.545i)10-s − 1.98·11-s + (5.73 + 1.21i)12-s + (2.72 + 4.72i)13-s + (−4.12 + 7.13i)14-s + (−1.90 + 2.11i)15-s + (0.276 − 2.62i)16-s + (3.60 + 3.99i)17-s + ⋯ |
| L(s) = 1 | + (−1.37 − 0.612i)2-s + (1.07 − 0.784i)3-s + (0.848 + 0.942i)4-s + (−0.538 + 0.114i)5-s + (−1.96 + 0.417i)6-s + (0.152 − 1.45i)7-s + (−0.124 − 0.383i)8-s + (0.240 − 0.741i)9-s + (0.811 + 0.172i)10-s − 0.598·11-s + (1.65 + 0.351i)12-s + (0.756 + 1.31i)13-s + (−1.10 + 1.90i)14-s + (−0.491 + 0.545i)15-s + (0.0690 − 0.656i)16-s + (0.873 + 0.969i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0529 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0529 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.439868 - 0.417149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.439868 - 0.417149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (-1.71 + 7.61i)T \) |
| good | 2 | \( 1 + (1.94 + 0.866i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-1.86 + 1.35i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (1.20 - 0.255i)T + (4.56 - 2.03i)T^{2} \) |
| 7 | \( 1 + (-0.404 + 3.84i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 + (-2.72 - 4.72i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.60 - 3.99i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.472 - 4.49i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.24 + 6.91i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.207 - 0.359i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.26 - 1.00i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (5.96 + 4.33i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.12 + 2.26i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.69 - 1.88i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.97 + 3.42i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.927 - 2.85i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.34 + 0.596i)T + (39.4 + 43.8i)T^{2} \) |
| 67 | \( 1 + (-5.93 + 1.26i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (8.53 + 1.81i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (6.01 + 1.27i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (6.05 - 6.72i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-4.66 - 2.07i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (6.45 - 4.68i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.08 + 1.82i)T + (64.9 - 72.0i)T^{2} \) |
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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
−14.49018270235165206447060342418, −13.77744671117416568051819281131, −12.44218923543144447269359123698, −11.02571268625451295851196412808, −10.18622537881418871059700195455, −8.697886784022207376009937019726, −7.927428921691800932174143206230, −7.09265564391078943221245453043, −3.67574299331716867330484624595, −1.64273496346213143749791481861,
3.11644245816151765207780776988, 5.48593257735637499154800776046, 7.61329065009617209465787767853, 8.457570954912678468559288424755, 9.187185543976640161571531244845, 10.16483539832654606169173609238, 11.64980663035445505133637678477, 13.30849608283979701867316418737, 14.97178019883833640886931304398, 15.68265387582191517873689641151
