L(s) = 1 | + (−0.110 + 0.191i)2-s + (−0.787 − 1.36i)3-s + (0.975 + 1.68i)4-s + (0.5 − 0.866i)5-s + 0.348·6-s + (2.03 − 1.69i)7-s − 0.873·8-s + (0.260 − 0.451i)9-s + (0.110 + 0.191i)10-s + (−2.66 − 4.62i)11-s + (1.53 − 2.65i)12-s − 6.75·13-s + (0.100 + 0.576i)14-s − 1.57·15-s + (−1.85 + 3.21i)16-s + (1.89 + 3.28i)17-s + ⋯ |
L(s) = 1 | + (−0.0781 + 0.135i)2-s + (−0.454 − 0.787i)3-s + (0.487 + 0.844i)4-s + (0.223 − 0.387i)5-s + 0.142·6-s + (0.767 − 0.640i)7-s − 0.308·8-s + (0.0869 − 0.150i)9-s + (0.0349 + 0.0605i)10-s + (−0.804 − 1.39i)11-s + (0.443 − 0.767i)12-s − 1.87·13-s + (0.0267 + 0.154i)14-s − 0.406·15-s + (−0.463 + 0.803i)16-s + (0.460 + 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.711944 - 0.953485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.711944 - 0.953485i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.03 + 1.69i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.110 - 0.191i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.787 + 1.36i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2.66 + 4.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.75T + 13T^{2} \) |
| 17 | \( 1 + (-1.89 - 3.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.41 + 5.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + (2.97 + 5.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.44 - 7.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.65T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + (-4.53 + 7.86i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.824 + 1.42i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.61 + 2.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.26 + 2.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.627 + 1.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.40T + 71T^{2} \) |
| 73 | \( 1 + (4.76 + 8.25i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.98 - 6.90i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 + (-6.59 + 11.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
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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
−10.10058665452641230227050280926, −8.966632629482178957701382909224, −7.939922298366445611923863902641, −7.53434697404228687766452434878, −6.72998664417232511830396406612, −5.66827484326143652711312333855, −4.73411026936572043591202706370, −3.35468400931029834175091521410, −2.16482092106745506585592471897, −0.60651766016095832898268630981,
1.88480935163126552782056527365, 2.66919662613640670461826987164, 4.60253073500170143027235363839, 5.17739930747017001153271424276, 5.73850044883455348853358110724, 7.32847543658183986542168259732, 7.55611245881670933725475891546, 9.319278718216781943511885386026, 9.904422523287430196013766726669, 10.35579908641242743411719068219