| L(s) = 1 | + (0.0675 + 1.41i)2-s + (0.681 + 1.70i)3-s + (−0.0160 + 0.00153i)4-s + (0.132 − 0.546i)5-s + (−2.36 + 1.08i)6-s + (2.36 − 1.17i)7-s + (0.400 + 2.78i)8-s + (−0.258 + 0.246i)9-s + (0.783 + 0.151i)10-s + (−5.46 − 0.260i)11-s + (−0.0135 − 0.0262i)12-s + (−3.18 − 2.76i)13-s + (1.83 + 3.27i)14-s + (1.01 − 0.146i)15-s + (−3.95 + 0.763i)16-s + (2.00 − 2.81i)17-s + ⋯ |
| L(s) = 1 | + (0.0477 + 1.00i)2-s + (0.393 + 0.982i)3-s + (−0.00801 + 0.000765i)4-s + (0.0592 − 0.244i)5-s + (−0.966 + 0.441i)6-s + (0.895 − 0.445i)7-s + (0.141 + 0.985i)8-s + (−0.0863 + 0.0823i)9-s + (0.247 + 0.0477i)10-s + (−1.64 − 0.0785i)11-s + (−0.00390 − 0.00757i)12-s + (−0.884 − 0.766i)13-s + (0.489 + 0.876i)14-s + (0.263 − 0.0378i)15-s + (−0.989 + 0.190i)16-s + (0.485 − 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.884964 + 1.13308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.884964 + 1.13308i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.36 + 1.17i)T \) |
| 23 | \( 1 + (4.58 + 1.41i)T \) |
| good | 2 | \( 1 + (-0.0675 - 1.41i)T + (-1.99 + 0.190i)T^{2} \) |
| 3 | \( 1 + (-0.681 - 1.70i)T + (-2.17 + 2.07i)T^{2} \) |
| 5 | \( 1 + (-0.132 + 0.546i)T + (-4.44 - 2.29i)T^{2} \) |
| 11 | \( 1 + (5.46 + 0.260i)T + (10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (3.18 + 2.76i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.00 + 2.81i)T + (-5.56 - 16.0i)T^{2} \) |
| 19 | \( 1 + (2.51 + 3.53i)T + (-6.21 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-3.66 - 8.02i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-2.37 - 3.01i)T + (-7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (1.88 + 1.97i)T + (-1.76 + 36.9i)T^{2} \) |
| 41 | \( 1 + (0.322 + 1.09i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.95 - 0.569i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (9.85 + 5.69i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.73 - 1.63i)T + (41.6 + 32.7i)T^{2} \) |
| 59 | \( 1 + (0.461 - 2.39i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (3.12 + 1.24i)T + (44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (5.57 - 10.8i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (6.13 - 3.94i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (1.25 + 13.1i)T + (-71.6 + 13.8i)T^{2} \) |
| 79 | \( 1 + (-2.77 + 0.959i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-11.4 - 3.35i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-9.02 - 7.09i)T + (20.9 + 86.4i)T^{2} \) |
| 97 | \( 1 + (1.24 - 0.366i)T + (81.6 - 52.4i)T^{2} \) |
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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
−13.50140678152572218808977727334, −12.23531042059484455227786379877, −10.73589917224579451385847886764, −10.30905835791723701622715868606, −8.800914279748822747392873575792, −7.931472185085633391804651623982, −7.04513696475538352159784698452, −5.20539245090189676669917522761, −4.83346112697645214729260295520, −2.78957808754961203802807920378,
1.88391959859922651964723193619, 2.61265757931663118713140825385, 4.56237025493283408911495915496, 6.25731454680139484145647250272, 7.63066059838072223930144377719, 8.166363441420690044751665845414, 9.943604776348644704512322078893, 10.63262286199046345626799866890, 11.85475339008191312920890942153, 12.41634933788821775438486641752
