L(s) = 1 | + (0.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (−2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 13-s − 1.99·15-s + (0.5 − 0.866i)19-s + (−2 − 1.73i)21-s + (0.500 + 0.866i)25-s − 0.999·27-s − 4·29-s + (−4.5 − 7.79i)31-s + (0.999 − 1.73i)33-s + (1.00 − 5.19i)35-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.277·13-s − 0.516·15-s + (0.114 − 0.198i)19-s + (−0.436 − 0.377i)21-s + (0.100 + 0.173i)25-s − 0.192·27-s − 0.742·29-s + (−0.808 − 1.39i)31-s + (0.174 − 0.301i)33-s + (0.169 − 0.878i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.408658863571527293190856625463, −8.622925224689990078637687944563, −7.65725667411586337291753654268, −6.94428271219848098841953180851, −6.00197520102407497620148519531, −5.15662579069428321961009531296, −3.83092180207419896372703433654, −3.26441095245092673564742696620, −2.31137945246950741745179691776, 0,
1.47128043266629261208765739997, 2.84873014442573848270599844740, 3.80820691287412543547877362802, 4.79925515410611158297109433282, 5.75994367144477926262725928795, 6.88981117483715539852989262365, 7.34306094790659998382162316382, 8.366009501478786564022377620215, 8.946021065453619476239263809227