L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)15-s − 1.73i·21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s − 0.999i·27-s + (−0.5 + 0.866i)29-s − 1.73·35-s + (−0.5 − 0.866i)41-s − 0.999·45-s + (−0.866 + 1.5i)47-s + (−1 − 1.73i)49-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)15-s − 1.73i·21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s − 0.999i·27-s + (−0.5 + 0.866i)29-s − 1.73·35-s + (−0.5 − 0.866i)41-s − 0.999·45-s + (−0.866 + 1.5i)47-s + (−1 − 1.73i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.681279322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.681279322\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−8.546564919844828829919345306141, −8.004814483703214879193056806685, −7.30596986151142023124428141902, −6.96281087686341717999921986945, −5.51988635997385261160774587765, −4.67820127715461474887788639533, −3.92548777695194507721642050241, −3.27236108783644115050752032575, −1.70702938926254345453074260019, −1.05167309494442332897635701643,
1.99795644573682382093650688303, 2.62351533429455454109864376841, 3.42716770290077940993653432234, 4.47822221018584133888104016738, 5.12352981424206365054936397851, 6.13955074586187396378085917121, 7.02056358084159516102458500366, 7.928378221921026891476503055201, 8.417766070010862834619965150998, 8.969118298666118757854638260889