L(s) = 1 | − 2-s + (0.866 + 0.5i)3-s + 4-s + (0.866 − 1.5i)5-s + (−0.866 − 0.5i)6-s − 8-s + (0.499 + 0.866i)9-s + (−0.866 + 1.5i)10-s + (0.866 + 0.5i)12-s + (1.5 − 0.866i)15-s + 16-s + (−0.499 − 0.866i)18-s + (0.866 + 1.5i)19-s + (0.866 − 1.5i)20-s + (0.5 − 0.866i)23-s + (−0.866 − 0.5i)24-s + ⋯ |
L(s) = 1 | − 2-s + (0.866 + 0.5i)3-s + 4-s + (0.866 − 1.5i)5-s + (−0.866 − 0.5i)6-s − 8-s + (0.499 + 0.866i)9-s + (−0.866 + 1.5i)10-s + (0.866 + 0.5i)12-s + (1.5 − 0.866i)15-s + 16-s + (−0.499 − 0.866i)18-s + (0.866 + 1.5i)19-s + (0.866 − 1.5i)20-s + (0.5 − 0.866i)23-s + (−0.866 − 0.5i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.381339472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381339472\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.974580754702294557430528350514, −8.106274863983535447951220987993, −7.75295161268803583067242832865, −6.57886958433010366035368021234, −5.66814827901407948103492125458, −5.02613504011627037139446590308, −4.02478695232244343387917314903, −2.95883535136583688738117303430, −1.94375408824720839235719744808, −1.21458692847495199011085181740,
1.32034395543754513661483588273, 2.34246982427643622087069820684, 2.88151665973368974295905410272, 3.56784654393501426486002407151, 5.26430270649537209195397563684, 6.28682503289757458251499592501, 6.75603483608755255501974197502, 7.38266419910835152361691571060, 7.86603733682174083121396316367, 9.039928217849024582493904890522