L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.5 + 2.59i)5-s + (2.5 − 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)10-s + (−1.5 − 2.59i)11-s + 5·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (−2.5 + 4.33i)19-s + 3·20-s − 3·22-s + (−1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.670 + 1.16i)5-s + (0.944 − 0.327i)7-s − 0.353·8-s + (0.474 + 0.821i)10-s + (−0.452 − 0.783i)11-s + 1.38·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.573 + 0.993i)19-s + 0.670·20-s − 0.639·22-s + (−0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.894273557\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.894273557\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−10.34220122028087596798656835515, −8.891226833385801878122406863794, −8.101876096637942012191884497065, −7.48532144492719564883978653468, −6.22620596868304138213600402021, −5.64218873523986947280019276118, −4.15045448100220269358907368681, −3.69469586173892563049173973377, −2.61358185863628775441037354243, −1.21312791672291179919044185279,
0.931380735495143725136401995693, 2.55290428672454911294736394191, 4.18169729592797655969431318762, 4.59042064991422698850656927457, 5.42642939739238448349896038694, 6.40692489972887638870109508637, 7.56458326776227314569478269088, 8.167158492048686555481550602543, 8.722733110802035391253650484361, 9.522504121358386028611687476099