| L(s) = 1 | + (3.35 − 0.592i)3-s + (1.47 − 1.23i)5-s + (0.111 − 0.193i)7-s + (2.47 − 0.899i)9-s + (0.796 + 1.37i)11-s + (0.784 + 0.138i)13-s + (4.21 − 5.02i)15-s + (6.30 + 2.29i)17-s + (−15.7 + 10.6i)19-s + (0.260 − 0.716i)21-s + (−24.2 − 20.3i)23-s + (−3.69 + 20.9i)25-s + (−18.8 + 10.8i)27-s + (4.58 + 12.5i)29-s + (−43.5 − 25.1i)31-s + ⋯ |
| L(s) = 1 | + (1.11 − 0.197i)3-s + (0.294 − 0.247i)5-s + (0.0159 − 0.0276i)7-s + (0.274 − 0.0999i)9-s + (0.0723 + 0.125i)11-s + (0.0603 + 0.0106i)13-s + (0.281 − 0.335i)15-s + (0.370 + 0.134i)17-s + (−0.829 + 0.558i)19-s + (0.0124 − 0.0341i)21-s + (−1.05 − 0.883i)23-s + (−0.147 + 0.838i)25-s + (−0.696 + 0.402i)27-s + (0.158 + 0.434i)29-s + (−1.40 − 0.810i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.69139 - 0.191652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69139 - 0.191652i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (15.7 - 10.6i)T \) |
| good | 3 | \( 1 + (-3.35 + 0.592i)T + (8.45 - 3.07i)T^{2} \) |
| 5 | \( 1 + (-1.47 + 1.23i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-0.111 + 0.193i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.796 - 1.37i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-0.784 - 0.138i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-6.30 - 2.29i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (24.2 + 20.3i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-4.58 - 12.5i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (43.5 + 25.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 6.71iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-63.1 + 11.1i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-10.7 + 9.02i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-49.0 + 17.8i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-52.5 + 62.6i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-2.86 + 7.87i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (36.5 + 30.6i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-19.6 - 53.8i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-19.1 - 22.8i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (10.9 + 62.3i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-1.13 + 0.199i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-39.0 + 67.6i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-50.7 - 8.94i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (45.2 - 124. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−14.29381643033966241258333416002, −13.26868848451601565834578925789, −12.33382158193218017665699356827, −10.78087933505610459670133714389, −9.479211499877078344073274140171, −8.531395900245020190795958524262, −7.47975234723267683252886228462, −5.80169991251151203348960552377, −3.88505454326503673447461093184, −2.15340846418180902141832170598,
2.44619698330147952312133240924, 3.95349951581415308875471673311, 5.91017456090514145264045708766, 7.52330703764667081838640962099, 8.684693853924235943010044903974, 9.620331943615524052595246313154, 10.82946322284605830182285727259, 12.24070767144560128158323368577, 13.56867793091991436583283872192, 14.25565102637157758760460058119
