L(s) = 1 | + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s + 0.999i·55-s + (0.866 + 0.5i)59-s + (0.5 + 0.866i)61-s + (−0.499 − 0.866i)65-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s + 0.999i·55-s + (0.866 + 0.5i)59-s + (0.5 + 0.866i)61-s + (−0.499 − 0.866i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308050187\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308050187\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.390937886570871985252176906990, −8.579167839866973821029247455050, −7.82520614483452988238121293680, −7.27583185136222156985575791669, −5.94774051497234796275808604794, −5.24248666134070464560400147146, −4.66259989987023571304349153822, −3.53293102406919621341779264794, −2.20187990991086008338375275662, −1.12131364256116508425898221565,
1.70005822714374503798060240629, 2.62627162503586019791877367352, 3.59078343140711757516828772445, 4.90457158059993175440515498001, 5.49776867271052208112105938889, 6.52232047004454338084035725251, 7.09118454118665959403419269479, 8.218622599647811835367795538002, 8.698173160389272440767811078940, 9.642000638939927148493646718178