| L(s) = 1 | + (−1.41 + 0.0995i)2-s + (1.52 + 2.28i)3-s + (1.98 − 0.280i)4-s + (−0.326 − 0.0649i)5-s + (−2.37 − 3.06i)6-s + (3.74 − 0.745i)7-s + (−2.76 + 0.593i)8-s + (−1.73 + 4.18i)9-s + (0.466 + 0.0590i)10-s + (−3.13 − 2.09i)11-s + (3.66 + 4.09i)12-s + (−2.87 + 2.87i)13-s + (−5.21 + 1.42i)14-s + (−0.349 − 0.843i)15-s + (3.84 − 1.11i)16-s + (4.11 + 0.311i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0704i)2-s + (0.880 + 1.31i)3-s + (0.990 − 0.140i)4-s + (−0.145 − 0.0290i)5-s + (−0.971 − 1.25i)6-s + (1.41 − 0.281i)7-s + (−0.977 + 0.209i)8-s + (−0.578 + 1.39i)9-s + (0.147 + 0.0186i)10-s + (−0.943 − 0.630i)11-s + (1.05 + 1.18i)12-s + (−0.797 + 0.797i)13-s + (−1.39 + 0.380i)14-s + (−0.0902 − 0.217i)15-s + (0.960 − 0.278i)16-s + (0.997 + 0.0754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.815701 + 0.528284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.815701 + 0.528284i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0995i)T \) |
| 17 | \( 1 + (-4.11 - 0.311i)T \) |
| good | 3 | \( 1 + (-1.52 - 2.28i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (0.326 + 0.0649i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-3.74 + 0.745i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (3.13 + 2.09i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (2.87 - 2.87i)T - 13iT^{2} \) |
| 19 | \( 1 + (-0.183 - 0.442i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.77 + 1.18i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.575 + 2.89i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (4.13 + 6.18i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-3.97 + 2.65i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.10 - 5.57i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.938 + 2.26i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-9.23 + 9.23i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.71 - 3.19i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.37 - 0.983i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.45 - 12.3i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + 13.0iT - 67T^{2} \) |
| 71 | \( 1 + (1.11 - 0.743i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.580 + 2.91i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (4.26 - 6.37i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (-3.77 + 1.56i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (9.04 - 9.04i)T - 89iT^{2} \) |
| 97 | \( 1 + (10.4 + 2.07i)T + (89.6 + 37.1i)T^{2} \) |
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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
−13.89767198934815443589252395468, −11.91626424975459476385159668645, −10.97584874082479959368329364502, −10.15442550522513899260852643558, −9.295401607677086418233496229374, −8.145096341376874188668867186968, −7.70147155404193981893719914257, −5.49197225198055707866624232545, −4.11122362128177175266174035976, −2.40357408795475331658673363811,
1.62394187639631965968033175031, 2.79111887772279580344808835373, 5.45364109261397873543765805764, 7.30645250482772985339507201736, 7.74761471962306065755474133136, 8.421985172919424774893770636904, 9.746858174629456497242890026716, 10.98259974345778900561395133247, 12.16462781608926034754318858008, 12.71835364833181374442108185171
