| L(s) = 1 | + (−0.847 + 0.0741i)2-s + (−0.737 − 0.344i)3-s + (−1.25 + 0.221i)4-s + (0.116 − 2.23i)5-s + (0.650 + 0.236i)6-s + (1.02 − 3.83i)7-s + (2.69 − 0.721i)8-s + (−1.50 − 1.79i)9-s + (0.0671 + 1.90i)10-s + (−2.45 + 4.25i)11-s + (1.00 + 0.269i)12-s + (−0.515 − 1.10i)13-s + (−0.586 + 3.32i)14-s + (−0.853 + 1.60i)15-s + (0.173 − 0.0630i)16-s + (−0.122 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.598 + 0.0524i)2-s + (−0.426 − 0.198i)3-s + (−0.628 + 0.110i)4-s + (0.0519 − 0.998i)5-s + (0.265 + 0.0966i)6-s + (0.388 − 1.45i)7-s + (0.951 − 0.254i)8-s + (−0.500 − 0.596i)9-s + (0.0212 + 0.600i)10-s + (−0.740 + 1.28i)11-s + (0.289 + 0.0776i)12-s + (−0.143 − 0.306i)13-s + (−0.156 + 0.889i)14-s + (−0.220 + 0.415i)15-s + (0.0433 − 0.0157i)16-s + (−0.0297 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0969 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0969 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.345934 - 0.381273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.345934 - 0.381273i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.116 + 2.23i)T \) |
| 19 | \( 1 + (-4.22 - 1.07i)T \) |
| good | 2 | \( 1 + (0.847 - 0.0741i)T + (1.96 - 0.347i)T^{2} \) |
| 3 | \( 1 + (0.737 + 0.344i)T + (1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (-1.02 + 3.83i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.45 - 4.25i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.515 + 1.10i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (0.122 + 1.40i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-1.23 - 0.867i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (1.63 - 1.37i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-8.01 + 4.62i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.74 + 6.74i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.783 - 2.15i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.769 - 1.09i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (4.96 + 0.434i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (-5.31 + 7.59i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-9.46 - 7.94i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.09 + 6.20i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.766 - 8.76i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (2.67 + 0.471i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.0649 - 0.139i)T + (-46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (1.57 - 0.574i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.747 + 0.200i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-3.86 - 1.40i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (9.84 - 0.860i)T + (95.5 - 16.8i)T^{2} \) |
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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
−13.47252524386037408095954748186, −12.80938820344925679963193523245, −11.59477361355725147961296260341, −10.17809738800295579142009036157, −9.460248998884028448960653406321, −8.059113661627962064652885978810, −7.27984656530231963472995904407, −5.24978550666640104190504521111, −4.22023216361773372848474724605, −0.846634703143571102609541564884,
2.74909384500762460235069895060, 5.06802488671774517830777942854, 6.00122738266492443726543665012, 7.939145902655386773439120560823, 8.729807556549917563065343142788, 10.02461346984452055312444308508, 11.01653991812196503255185868646, 11.74219562822482550617499525221, 13.45397813036344810767834585992, 14.21612338923795054493685881843
