L(s) = 1 | + (−0.866 + 1.5i)5-s + (−0.5 + 0.866i)13-s − 1.73·17-s + (−1 − 1.73i)25-s + (0.866 + 1.5i)29-s − 37-s + (−0.5 + 0.866i)49-s + (0.5 + 0.866i)61-s + (−0.866 − 1.5i)65-s + 73-s + (1.49 − 2.59i)85-s + 1.73·89-s + (1 + 1.73i)97-s + 109-s + (0.866 − 1.5i)113-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.5i)5-s + (−0.5 + 0.866i)13-s − 1.73·17-s + (−1 − 1.73i)25-s + (0.866 + 1.5i)29-s − 37-s + (−0.5 + 0.866i)49-s + (0.5 + 0.866i)61-s + (−0.866 − 1.5i)65-s + 73-s + (1.49 − 2.59i)85-s + 1.73·89-s + (1 + 1.73i)97-s + 109-s + (0.866 − 1.5i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6381682878\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6381682878\) |
\(L(1)\) |
|
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 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−10.41793029983317460263587001971, −9.286983690778750795960351663191, −8.523071616805651160533726038411, −7.50413938217423730809569488293, −6.82678797575246410266089204738, −6.42978097528950900174302718348, −4.91118205042220952602071082669, −4.05091185198620093522574131547, −3.09721079523765228195275203066, −2.14564584167522378567864343335,
0.52240757343078394694957486898, 2.15186777005627048208888998857, 3.60036475798354218080354546801, 4.57977859764725086747004037325, 5.04473620360481570901459270148, 6.20230695646714002540597864779, 7.27080429555629325172474980546, 8.161390574520834560190547474342, 8.593591867892220266455625611607, 9.411317547906686214372234070467