L(s) = 1 | + 1.73·3-s + (1.5 + 0.866i)5-s + (−2.59 + 1.5i)7-s + 2.99·9-s + (−2.59 − 4.5i)11-s + (−0.5 + 0.866i)13-s + (2.59 + 1.49i)15-s − 3.46i·17-s + 6i·19-s + (−4.5 + 2.59i)21-s + (2.59 − 4.5i)23-s + (−1 − 1.73i)25-s + 5.19·27-s + (−7.5 + 4.33i)29-s + (−2.59 − 1.5i)31-s + ⋯ |
L(s) = 1 | + 1.00·3-s + (0.670 + 0.387i)5-s + (−0.981 + 0.566i)7-s + 0.999·9-s + (−0.783 − 1.35i)11-s + (−0.138 + 0.240i)13-s + (0.670 + 0.387i)15-s − 0.840i·17-s + 1.37i·19-s + (−0.981 + 0.566i)21-s + (0.541 − 0.938i)23-s + (−0.200 − 0.346i)25-s + 1.00·27-s + (−1.39 + 0.804i)29-s + (−0.466 − 0.269i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44601 + 0.126509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44601 + 0.126509i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 + 4.5i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (-2.59 + 4.5i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.5 - 4.33i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.59 + 1.5i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.59 - 1.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.59 - 4.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (-12.9 + 7.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.59 - 4.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.46iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.25644918695345062455243655222, −12.45553668997881141967631147022, −10.88113874778016636316576246470, −9.861275374708956164120571079545, −9.076031765902543733084110168792, −8.008019579604134389172515308716, −6.65780360780015032693371357846, −5.54237733388065670401573669277, −3.48429164753583361322936142108, −2.46632656362164108024302273549,
2.13731651477474007062868200648, 3.69268206051162075448026450011, 5.17852515331783687498942647739, 6.87550645366045446450862709724, 7.71201406813773082747995185529, 9.247501654964926554511061083515, 9.694107273038700793747814262090, 10.72482071138261257623531497156, 12.64312149583815458770661577064, 13.09496732652307132032936546049