| L(s) = 1 | + (0.642 + 0.766i)2-s + (1.43 + 1.20i)3-s + (−0.173 + 0.984i)4-s + (−1.45 − 4.00i)5-s + 1.87i·6-s + (−3.39 + 1.23i)7-s + (−0.866 + 0.500i)8-s + (0.0923 + 0.524i)9-s + (2.13 − 3.69i)10-s + (1.05 + 1.83i)11-s + (−1.43 + 1.20i)12-s + (2.84 + 0.500i)13-s + (−3.12 − 1.80i)14-s + (2.74 − 7.53i)15-s + (−0.939 − 0.342i)16-s + (0.0263 − 0.00463i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (0.831 + 0.697i)3-s + (−0.0868 + 0.492i)4-s + (−0.652 − 1.79i)5-s + 0.767i·6-s + (−1.28 + 0.466i)7-s + (−0.306 + 0.176i)8-s + (0.0307 + 0.174i)9-s + (0.674 − 1.16i)10-s + (0.319 + 0.552i)11-s + (−0.415 + 0.348i)12-s + (0.787 + 0.138i)13-s + (−0.835 − 0.482i)14-s + (0.707 − 1.94i)15-s + (−0.234 − 0.0855i)16-s + (0.00638 − 0.00112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11708 + 0.452583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11708 + 0.452583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 37 | \( 1 + (4.49 - 4.09i)T \) |
| good | 3 | \( 1 + (-1.43 - 1.20i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (1.45 + 4.00i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.39 - 1.23i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.05 - 1.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.84 - 0.500i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0263 + 0.00463i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (2.07 - 2.47i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.57 - 1.48i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.96 + 2.86i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.76iT - 31T^{2} \) |
| 41 | \( 1 + (-0.259 + 1.46i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 5.53iT - 43T^{2} \) |
| 47 | \( 1 + (1.30 - 2.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.79 + 0.652i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.92 + 5.28i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (5.65 + 0.996i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.50 - 2.36i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (7.10 + 5.96i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + (0.484 + 1.32i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.294 + 1.67i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.82 + 7.77i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.65 - 3.26i)T + (48.5 + 84.0i)T^{2} \) |
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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
−15.05683259438011521656808106087, −13.56014237543109086876126528752, −12.71532386091061283943739215057, −11.92303541377000272756020418805, −9.650959651129666869425297339778, −8.975042040887139885815403727142, −8.090235817728825550914116417678, −6.18731652692186475958159454041, −4.57894080905580908287929945523, −3.56608600280887833588013475599,
2.85748680373103714221288870135, 3.57366065209058556957467984464, 6.46696189581166297350921686575, 7.12413297579495938977416186534, 8.687196328610354018412793528136, 10.37943377493979445805594529191, 11.01709147654437644300796222586, 12.44407441174347952798640188068, 13.59858349823213489218579303670, 14.11011426200744505690565111232
