| 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
−14.25565102637157758760460058119, −13.56867793091991436583283872192, −12.24070767144560128158323368577, −10.82946322284605830182285727259, −9.620331943615524052595246313154, −8.684693853924235943010044903974, −7.52330703764667081838640962099, −5.91017456090514145264045708766, −3.95349951581415308875471673311, −2.44619698330147952312133240924,
2.15340846418180902141832170598, 3.88505454326503673447461093184, 5.80169991251151203348960552377, 7.47975234723267683252886228462, 8.531395900245020190795958524262, 9.479211499877078344073274140171, 10.78087933505610459670133714389, 12.33382158193218017665699356827, 13.26868848451601565834578925789, 14.29381643033966241258333416002
