| L(s) = 1 | + (−0.660 + 2.37i)2-s + (−1.11 + 1.32i)3-s + (−3.51 − 2.11i)4-s + (−0.952 − 0.0516i)5-s + (−2.41 − 3.53i)6-s + (−0.387 + 2.36i)7-s + (3.76 − 3.56i)8-s + (−0.507 − 2.95i)9-s + (0.752 − 2.23i)10-s + (1.09 − 2.05i)11-s + (6.72 − 2.29i)12-s + (−4.15 + 1.65i)13-s + (−5.37 − 2.48i)14-s + (1.13 − 1.20i)15-s + (2.15 + 4.06i)16-s + (1.79 − 0.294i)17-s + ⋯ |
| L(s) = 1 | + (−0.467 + 1.68i)2-s + (−0.644 + 0.764i)3-s + (−1.75 − 1.05i)4-s + (−0.426 − 0.0231i)5-s + (−0.985 − 1.44i)6-s + (−0.146 + 0.893i)7-s + (1.33 − 1.26i)8-s + (−0.169 − 0.985i)9-s + (0.237 − 0.706i)10-s + (0.328 − 0.619i)11-s + (1.93 − 0.661i)12-s + (−1.15 + 0.458i)13-s + (−1.43 − 0.664i)14-s + (0.292 − 0.310i)15-s + (0.539 + 1.01i)16-s + (0.436 − 0.0715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.229907 - 0.254866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.229907 - 0.254866i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.11 - 1.32i)T \) |
| 59 | \( 1 + (-1.00 - 7.61i)T \) |
| good | 2 | \( 1 + (0.660 - 2.37i)T + (-1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (0.952 + 0.0516i)T + (4.97 + 0.540i)T^{2} \) |
| 7 | \( 1 + (0.387 - 2.36i)T + (-6.63 - 2.23i)T^{2} \) |
| 11 | \( 1 + (-1.09 + 2.05i)T + (-6.17 - 9.10i)T^{2} \) |
| 13 | \( 1 + (4.15 - 1.65i)T + (9.43 - 8.94i)T^{2} \) |
| 17 | \( 1 + (-1.79 + 0.294i)T + (16.1 - 5.42i)T^{2} \) |
| 19 | \( 1 + (-3.93 - 4.63i)T + (-3.07 + 18.7i)T^{2} \) |
| 23 | \( 1 + (4.81 - 3.65i)T + (6.15 - 22.1i)T^{2} \) |
| 29 | \( 1 + (2.94 - 0.816i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (7.25 + 6.16i)T + (5.01 + 30.5i)T^{2} \) |
| 37 | \( 1 + (5.20 - 5.49i)T + (-2.00 - 36.9i)T^{2} \) |
| 41 | \( 1 + (3.48 - 4.57i)T + (-10.9 - 39.5i)T^{2} \) |
| 43 | \( 1 + (-7.16 + 3.79i)T + (24.1 - 35.5i)T^{2} \) |
| 47 | \( 1 + (-0.235 - 4.34i)T + (-46.7 + 5.08i)T^{2} \) |
| 53 | \( 1 + (0.628 + 1.86i)T + (-42.1 + 32.0i)T^{2} \) |
| 61 | \( 1 + (-5.01 - 1.39i)T + (52.2 + 31.4i)T^{2} \) |
| 67 | \( 1 + (-5.10 - 5.38i)T + (-3.62 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-12.6 + 0.684i)T + (70.5 - 7.67i)T^{2} \) |
| 73 | \( 1 + (4.73 - 10.2i)T + (-47.2 - 55.6i)T^{2} \) |
| 79 | \( 1 + (3.58 - 5.28i)T + (-29.2 - 73.3i)T^{2} \) |
| 83 | \( 1 + (-13.7 - 3.02i)T + (75.3 + 34.8i)T^{2} \) |
| 89 | \( 1 + (0.970 + 3.49i)T + (-76.2 + 45.8i)T^{2} \) |
| 97 | \( 1 + (0.0787 + 0.170i)T + (-62.7 + 73.9i)T^{2} \) |
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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
−13.96188654917545150598435853673, −12.22648294329362838986373408188, −11.53807236756921932182443273763, −9.834635235084346735957223975360, −9.399594604971538722712643969119, −8.218011721020611043029454445686, −7.20502617613885471819833313223, −5.85473076057034287769772845849, −5.42504533568667425149152297266, −3.91131411487509240476422252253,
0.39168649571161801383629773457, 2.09087958131050214109849316700, 3.72239211719657736421723093111, 5.06554763743099717589930266384, 7.07188427033447618260647709075, 7.83556442087262048262662465296, 9.344454336921010999612130555505, 10.30698230105778734391810169493, 11.00697405948589238579644135754, 12.07588986934084054588872569514
