L(s) = 1 | + (−0.876 − 0.563i)2-s + (0.959 + 0.281i)3-s + (−0.379 − 0.831i)4-s + (2.49 − 2.88i)5-s + (−0.682 − 0.787i)6-s + (−1.63 − 1.05i)7-s + (−0.432 + 3.00i)8-s + (0.841 + 0.540i)9-s + (−3.81 + 1.12i)10-s + (−2.53 + 2.92i)11-s + (−0.130 − 0.904i)12-s + (−0.695 − 4.83i)13-s + (0.843 + 1.84i)14-s + (3.20 − 2.06i)15-s + (0.876 − 1.01i)16-s + (−0.324 + 0.711i)17-s + ⋯ |
L(s) = 1 | + (−0.620 − 0.398i)2-s + (0.553 + 0.162i)3-s + (−0.189 − 0.415i)4-s + (1.11 − 1.28i)5-s + (−0.278 − 0.321i)6-s + (−0.619 − 0.398i)7-s + (−0.152 + 1.06i)8-s + (0.280 + 0.180i)9-s + (−1.20 + 0.354i)10-s + (−0.764 + 0.881i)11-s + (−0.0375 − 0.261i)12-s + (−0.192 − 1.34i)13-s + (0.225 + 0.493i)14-s + (0.828 − 0.532i)15-s + (0.219 − 0.252i)16-s + (−0.0787 + 0.172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0684 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0684 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.712800 - 0.763350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.712800 - 0.763350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (7.76 + 2.58i)T \) |
good | 2 | \( 1 + (0.876 + 0.563i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-2.49 + 2.88i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (1.63 + 1.05i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (2.53 - 2.92i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.695 + 4.83i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.324 - 0.711i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-5.08 + 3.26i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (-6.62 - 1.94i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 + 3.48T + 29T^{2} \) |
| 31 | \( 1 + (1.21 - 8.43i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + 0.994T + 37T^{2} \) |
| 41 | \( 1 + (1.64 - 3.60i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.634 - 1.38i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-12.9 - 3.79i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (-3.39 - 7.43i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.588 + 4.09i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-4.57 - 5.28i)T + (-8.68 + 60.3i)T^{2} \) |
| 71 | \( 1 + (0.548 + 1.20i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.14 - 3.62i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.272 - 1.89i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (6.87 - 7.93i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (4.49 - 1.31i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−12.47661483507067095305986850302, −10.73178882958525116623617612648, −9.971201971834199777940833615723, −9.391957946907400935568935504071, −8.632560791028450506504317154029, −7.35664650030720621408602056060, −5.50106888914796699488434622298, −4.94158964385567455620385876737, −2.75303808162731587489533155932, −1.16857484460813587576879150456,
2.49634525521537074953075288068, 3.51695797605033068893949971989, 5.76017585532329190756574866912, 6.80515010479751909597643328680, 7.52753077163266033123856486243, 8.937662670307044294796378914753, 9.493213715818892810528242488208, 10.40906844333989788391572054292, 11.66837545831680590392898470694, 13.04772254398514911945439620997