| L(s) = 1 | + (0.984 + 0.173i)2-s + (−0.266 − 1.50i)3-s + (0.939 + 0.342i)4-s + (−0.247 − 0.294i)5-s − 1.53i·6-s + (−2.50 + 2.09i)7-s + (0.866 + 0.5i)8-s + (0.613 − 0.223i)9-s + (−0.192 − 0.333i)10-s + (−1.29 + 2.23i)11-s + (0.266 − 1.50i)12-s + (−0.466 + 1.28i)13-s + (−2.82 + 1.63i)14-s + (−0.378 + 0.451i)15-s + (0.766 + 0.642i)16-s + (−1.13 − 3.13i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (−0.153 − 0.871i)3-s + (0.469 + 0.171i)4-s + (−0.110 − 0.131i)5-s − 0.625i·6-s + (−0.945 + 0.793i)7-s + (0.306 + 0.176i)8-s + (0.204 − 0.0744i)9-s + (−0.0608 − 0.105i)10-s + (−0.389 + 0.674i)11-s + (0.0768 − 0.435i)12-s + (−0.129 + 0.355i)13-s + (−0.755 + 0.436i)14-s + (−0.0978 + 0.116i)15-s + (0.191 + 0.160i)16-s + (−0.276 − 0.759i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16844 - 0.201005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16844 - 0.201005i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 37 | \( 1 + (-0.543 - 6.05i)T \) |
| good | 3 | \( 1 + (0.266 + 1.50i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (0.247 + 0.294i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.50 - 2.09i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (1.29 - 2.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.466 - 1.28i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.13 + 3.13i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (5.89 - 1.03i)T + (17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-6.53 + 3.77i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.78 - 1.60i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 41 | \( 1 + (-7.77 - 2.82i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + 4.33iT - 43T^{2} \) |
| 47 | \( 1 + (2.61 + 4.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.64 + 5.57i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-8.33 + 9.93i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.39 - 6.58i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (8.60 - 7.22i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.60 - 14.7i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + (-0.940 - 1.12i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.0104 + 0.00380i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.612 - 0.730i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-12.8 + 7.40i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.47797937456682821505776411752, −12.95776550460385992062539728262, −12.74123999058868198354530428339, −11.72657930687632417779776821824, −10.13053216161268090988534933826, −8.650453428374130119843433836975, −7.04874811397570534424476070680, −6.33084835150046139026160270806, −4.65317876991440684369376646399, −2.54674771238042052571562338493,
3.32864567113728287034536323364, 4.49469237045594641533512962211, 6.03380888885361611748833575069, 7.40711346034533821485602642102, 9.270724970515713181181972652884, 10.55725741757128463174548186183, 10.98799460386923159429163749379, 12.85443667651532735326832607887, 13.30756476310710664278851592650, 14.81391206304788343329366694347
