| L(s) = 1 | + (−0.766 + 0.642i)2-s + (2.55 + 0.931i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−2.55 + 0.931i)6-s + (2.33 − 4.04i)7-s + (0.500 + 0.866i)8-s + (3.37 + 2.83i)9-s + (0.766 + 0.642i)10-s + (0.0690 + 0.119i)11-s + (1.36 − 2.35i)12-s + (−3.59 + 1.30i)13-s + (0.810 + 4.59i)14-s + (0.472 − 2.68i)15-s + (−0.939 − 0.342i)16-s + (−4.53 + 3.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (1.47 + 0.537i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (−1.04 + 0.380i)6-s + (0.882 − 1.52i)7-s + (0.176 + 0.306i)8-s + (1.12 + 0.945i)9-s + (0.242 + 0.203i)10-s + (0.0208 + 0.0360i)11-s + (0.392 − 0.680i)12-s + (−0.996 + 0.362i)13-s + (0.216 + 1.22i)14-s + (0.122 − 0.692i)15-s + (−0.234 − 0.0855i)16-s + (−1.09 + 0.922i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38937 + 0.290070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38937 + 0.290070i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (3.96 - 1.80i)T \) |
| good | 3 | \( 1 + (-2.55 - 0.931i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-2.33 + 4.04i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0690 - 0.119i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.59 - 1.30i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (4.53 - 3.80i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.04 - 5.93i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.90 - 4.11i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.80 + 4.86i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.34T + 37T^{2} \) |
| 41 | \( 1 + (1.71 + 0.622i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.14 - 6.49i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.39 + 4.52i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.642 + 3.64i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.81 + 5.71i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 9.00i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.35 - 2.81i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.856 - 4.85i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (9.44 + 3.43i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-13.0 - 4.76i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.98 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 4.28i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.18 + 4.35i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.02630878380024131377661202513, −11.31427673563183340905808503684, −10.26688472573216744898466801301, −9.554112270716929967912856597623, −8.381615481233418101086895558391, −7.933750844823563369898502124677, −6.83414343016776104727744300806, −4.72579306267467678277183400859, −3.93990660438385033226928591239, −1.89263850793778430271031819893,
2.34130229118987550840330848831, 2.63885517039357644488500378530, 4.66992680828509352681808777427, 6.63056759634597898854405013361, 7.79639390827523068497843499620, 8.602710085577142760474857966044, 9.116824718046015679765272139679, 10.39572334535333035406819241979, 11.67443718256080468858779530897, 12.38334906745621306574836843022
