| L(s) = 1 | + (1.43 + 0.524i)2-s + (−0.173 − 0.984i)3-s + (0.266 + 0.223i)4-s + (−1.93 + 1.62i)5-s + (0.266 − 1.50i)6-s + (−0.266 + 0.460i)7-s + (−1.26 − 2.19i)8-s + (−0.939 + 0.342i)9-s + (−3.64 + 1.32i)10-s + (2.55 + 4.42i)11-s + (0.173 − 0.300i)12-s + (0.705 − 4.00i)13-s + (−0.624 + 0.524i)14-s + (1.93 + 1.62i)15-s + (−0.794 − 4.50i)16-s + (−1.82 − 0.664i)17-s + ⋯ |
| L(s) = 1 | + (1.01 + 0.370i)2-s + (−0.100 − 0.568i)3-s + (0.133 + 0.111i)4-s + (−0.867 + 0.727i)5-s + (0.108 − 0.615i)6-s + (−0.100 + 0.174i)7-s + (−0.447 − 0.775i)8-s + (−0.313 + 0.114i)9-s + (−1.15 + 0.419i)10-s + (0.769 + 1.33i)11-s + (0.0501 − 0.0868i)12-s + (0.195 − 1.11i)13-s + (−0.166 + 0.140i)14-s + (0.500 + 0.420i)15-s + (−0.198 − 1.12i)16-s + (−0.442 − 0.161i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12479 + 0.0681668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12479 + 0.0681668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-2.23 + 3.74i)T \) |
| good | 2 | \( 1 + (-1.43 - 0.524i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (1.93 - 1.62i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.266 - 0.460i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.55 - 4.42i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.705 + 4.00i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.82 + 0.664i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.33 - 1.95i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.51 - 0.550i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.93 - 8.55i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + (1.47 + 8.34i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.135 - 0.113i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.10 - 2.58i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (7.58 + 6.36i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.58 - 1.30i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (10.6 + 8.95i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.59 + 3.49i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.92 + 2.45i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.322 + 1.82i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.82 - 10.3i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.73 + 9.93i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.02 - 5.83i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.39 - 2.32i)T + (74.3 + 62.3i)T^{2} \) |
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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
−15.13384740648280833474300574341, −14.27618193468874764266538236534, −13.00986824282693871185425910569, −12.23599101053519380626316007286, −11.03239915415556385887399919309, −9.357700394794782790202010497145, −7.45943078410886197661341543102, −6.64206074559130170624967366671, −5.02236294594966168153223302238, −3.39060129821530087042324845572,
3.63485472156776633607676744889, 4.46851647850660291802023112003, 6.03447793611712914005906134233, 8.249402599013603649521214469168, 9.239237373008189523580833301756, 11.27854556235456173823790131903, 11.71279585280466856738439284399, 12.99479887362726887944844204998, 14.03025659970480124915438553699, 14.98843911517951175334623654254
