L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + (0.5 − 2.59i)7-s + (0.499 − 0.866i)9-s + (0.866 − 0.5i)11-s + 2i·13-s − 0.999·15-s + (2 + 3.46i)17-s + (0.866 + 2.5i)21-s + (3 − 5.19i)23-s + (−2 − 3.46i)25-s + 0.999i·27-s − i·29-s + (−2.5 − 4.33i)31-s + (−0.499 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.288i)3-s + (0.387 + 0.223i)5-s + (0.188 − 0.981i)7-s + (0.166 − 0.288i)9-s + (0.261 − 0.150i)11-s + 0.554i·13-s − 0.258·15-s + (0.485 + 0.840i)17-s + (0.188 + 0.545i)21-s + (0.625 − 1.08i)23-s + (−0.400 − 0.692i)25-s + 0.192i·27-s − 0.185i·29-s + (−0.449 − 0.777i)31-s + (−0.0870 + 0.150i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.544875251\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.544875251\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-0.866 - 0.5i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + iT - 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.92 - 4i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−9.879937511283278420013385829500, −8.774590751200133494742800654636, −7.942871599186222549862329000988, −6.90870889708230638409358298091, −6.35075993104240429749694346757, −5.39886923316976838175778589649, −4.34112089325055903693800452295, −3.72193247310207181514658953347, −2.23026642879936309966717864266, −0.828171625430901506055127439959,
1.14932013606313139914859897354, 2.34004166418688148155780056657, 3.47259113120174204283952077737, 4.95466111513168016948237441779, 5.45164368875705629741803709147, 6.19419504212507093691634602437, 7.27292738613870357373500518646, 7.923613413346811433083758210429, 9.111524840181189145191509581173, 9.429066892289935571875063212813