L(s) = 1 | + (0.796 − 0.605i)2-s + (−0.468 + 0.883i)3-s + (0.267 − 0.963i)4-s + (0.697 − 0.660i)5-s + (0.161 + 0.986i)6-s + (−1.30 − 1.53i)7-s + (−0.370 − 0.928i)8-s + (−0.561 − 0.827i)9-s + (0.155 − 0.947i)10-s + (4.44 − 2.67i)11-s + (0.725 + 0.687i)12-s + (2.29 − 3.37i)13-s + (−1.97 − 0.434i)14-s + (0.256 + 0.925i)15-s + (−0.856 − 0.515i)16-s + (−2.44 + 2.87i)17-s + ⋯ |
L(s) = 1 | + (0.562 − 0.427i)2-s + (−0.270 + 0.510i)3-s + (0.133 − 0.481i)4-s + (0.311 − 0.295i)5-s + (0.0660 + 0.402i)6-s + (−0.494 − 0.581i)7-s + (−0.130 − 0.328i)8-s + (−0.187 − 0.275i)9-s + (0.0491 − 0.299i)10-s + (1.34 − 0.806i)11-s + (0.209 + 0.198i)12-s + (0.635 − 0.937i)13-s + (−0.527 − 0.116i)14-s + (0.0663 + 0.238i)15-s + (−0.214 − 0.128i)16-s + (−0.593 + 0.698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53079 - 0.816760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53079 - 0.816760i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.796 + 0.605i)T \) |
| 3 | \( 1 + (0.468 - 0.883i)T \) |
| 59 | \( 1 + (7.04 - 3.05i)T \) |
good | 5 | \( 1 + (-0.697 + 0.660i)T + (0.270 - 4.99i)T^{2} \) |
| 7 | \( 1 + (1.30 + 1.53i)T + (-1.13 + 6.90i)T^{2} \) |
| 11 | \( 1 + (-4.44 + 2.67i)T + (5.15 - 9.71i)T^{2} \) |
| 13 | \( 1 + (-2.29 + 3.37i)T + (-4.81 - 12.0i)T^{2} \) |
| 17 | \( 1 + (2.44 - 2.87i)T + (-2.75 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-6.22 - 2.87i)T + (12.3 + 14.4i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 0.716i)T + (18.3 - 13.9i)T^{2} \) |
| 29 | \( 1 + (6.35 + 4.83i)T + (7.75 + 27.9i)T^{2} \) |
| 31 | \( 1 + (8.42 - 3.89i)T + (20.0 - 23.6i)T^{2} \) |
| 37 | \( 1 + (-1.87 + 4.71i)T + (-26.8 - 25.4i)T^{2} \) |
| 41 | \( 1 + (10.1 + 3.43i)T + (32.6 + 24.8i)T^{2} \) |
| 43 | \( 1 + (-10.9 - 6.55i)T + (20.1 + 37.9i)T^{2} \) |
| 47 | \( 1 + (-4.18 - 3.96i)T + (2.54 + 46.9i)T^{2} \) |
| 53 | \( 1 + (-0.601 - 3.66i)T + (-50.2 + 16.9i)T^{2} \) |
| 61 | \( 1 + (9.80 - 7.45i)T + (16.3 - 58.7i)T^{2} \) |
| 67 | \( 1 + (-4.03 - 10.1i)T + (-48.6 + 46.0i)T^{2} \) |
| 71 | \( 1 + (6.41 + 6.07i)T + (3.84 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-6.37 - 1.40i)T + (66.2 + 30.6i)T^{2} \) |
| 79 | \( 1 + (-5.15 - 9.73i)T + (-44.3 + 65.3i)T^{2} \) |
| 83 | \( 1 + (2.12 + 0.231i)T + (81.0 + 17.8i)T^{2} \) |
| 89 | \( 1 + (-8.71 - 6.62i)T + (23.8 + 85.7i)T^{2} \) |
| 97 | \( 1 + (-0.915 + 0.201i)T + (88.0 - 40.7i)T^{2} \) |
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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
−11.17744181855607104207033246358, −10.67389649665929524218004916976, −9.535625270156219383668722153529, −8.927120747043069749614111974576, −7.38285395893995495018708189113, −6.07284876163421763006101025090, −5.51833784182615249997091356545, −3.96532466563473445355991313283, −3.39855960965552789897758422843, −1.21481904543481366279174137604,
1.94359247921457287421211396065, 3.45108614093424650789323183586, 4.80071373596143556038219146316, 5.96634716010650091950263263003, 6.77690100182168212999491269276, 7.35545110931400123725986970910, 9.065619896073751430991957105688, 9.383659629179231621587934270286, 11.07710256415695482193518227366, 11.77335336228340876106452648708