L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + i·7-s + (−0.499 + 0.866i)9-s + (0.866 − 0.5i)11-s − 0.999·12-s + (−0.866 − 0.5i)13-s + (−0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)19-s − 0.999·20-s + (−0.866 + 0.5i)21-s + 23-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + i·7-s + (−0.499 + 0.866i)9-s + (0.866 − 0.5i)11-s − 0.999·12-s + (−0.866 − 0.5i)13-s + (−0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)19-s − 0.999·20-s + (−0.866 + 0.5i)21-s + 23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.232144586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.232144586\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 - iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.865944661642601476919361237520, −9.144579819720679780490670821840, −8.951933743984817935298252684505, −7.68292453288654847516603239886, −7.14868866823476633364515473715, −5.65979299599927104035211279314, −5.18015505353844336756382200541, −3.86638498920024900584717128510, −3.07539682180112874128978521843, −2.47020962831691831250803947116,
1.16905796660583495902150279351, 1.71326617048426051152931158465, 3.47872133900900992813000905752, 4.52355382343400520020966575399, 5.37990334083587375773434503036, 6.31088634539373859357618005381, 7.16589398490219261997396560826, 7.896217400761290188464828547643, 8.977911122570370094087615637513, 9.562991585285932689141769927222