| L(s) = 1 | + (−1.46 − 1.83i)2-s + (7.38 − 3.55i)3-s + (0.557 − 2.44i)4-s + (−0.458 + 0.220i)5-s + (−17.3 − 8.33i)6-s + (12.0 + 14.0i)7-s + (−22.1 + 10.6i)8-s + (25.0 − 31.4i)9-s + (1.07 + 0.517i)10-s + (−23.3 − 29.2i)11-s + (−4.57 − 20.0i)12-s + (−8.07 − 10.1i)13-s + (8.22 − 42.6i)14-s + (−2.59 + 3.25i)15-s + (33.9 + 16.3i)16-s + (27.8 + 121. i)17-s + ⋯ |
| L(s) = 1 | + (−0.516 − 0.647i)2-s + (1.42 − 0.684i)3-s + (0.0697 − 0.305i)4-s + (−0.0409 + 0.0197i)5-s + (−1.17 − 0.567i)6-s + (0.649 + 0.760i)7-s + (−0.980 + 0.472i)8-s + (0.927 − 1.16i)9-s + (0.0339 + 0.0163i)10-s + (−0.639 − 0.802i)11-s + (−0.109 − 0.481i)12-s + (−0.172 − 0.215i)13-s + (0.157 − 0.813i)14-s + (−0.0447 + 0.0560i)15-s + (0.530 + 0.255i)16-s + (0.397 + 1.73i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0288 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.11014 - 1.14263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11014 - 1.14263i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-12.0 - 14.0i)T \) |
| good | 2 | \( 1 + (1.46 + 1.83i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (-7.38 + 3.55i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (0.458 - 0.220i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (23.3 + 29.2i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (8.07 + 10.1i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-27.8 - 121. i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 - 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-2.99 + 13.1i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (23.4 + 102. i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 - 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (21.6 + 94.9i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (160. - 77.4i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-379. - 182. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (387. + 486. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (21.6 - 95.0i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (583. + 281. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (84.7 + 371. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 + 418.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (150. - 661. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-78.3 + 98.2i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + 952.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (223. - 280. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-108. + 136. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + 1.35e3T + 9.12e5T^{2} \) |
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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
−14.77996130840363761847498103008, −13.74409406109212997694440180342, −12.50771704863125030363189883550, −11.24301068128342704795434183757, −9.857257919613049007908131707738, −8.605712579625655091577316242966, −7.949333277300743681355560827533, −5.82662500309185223979561463128, −3.01798884800993934627923002024, −1.67982775256840169900647552129,
2.93083790453382595545766685292, 4.58410282676039513842527339073, 7.31753469059038692200410236966, 7.934111801160523588341069701323, 9.207428807688645091172698991474, 10.10318329163252445317493267902, 11.93265796481855455408223712334, 13.60064478848832953369905467849, 14.42237679004099146488721287438, 15.55994999380371640525170918963
