| L(s) = 1 | + (−7.74 + 13.4i)2-s + (−55.9 − 96.9i)4-s + (−52.7 − 91.3i)5-s + (761. − 1.31e3i)7-s − 249.·8-s + 1.63e3·10-s + (1.04e3 − 1.81e3i)11-s + (4.66e3 + 8.08e3i)13-s + (1.17e4 + 2.04e4i)14-s + (9.09e3 − 1.57e4i)16-s + 2.06e4·17-s − 9.36e3·19-s + (−5.90e3 + 1.02e4i)20-s + (1.61e4 + 2.80e4i)22-s + (−3.86e4 − 6.70e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.684 + 1.18i)2-s + (−0.437 − 0.757i)4-s + (−0.188 − 0.326i)5-s + (0.839 − 1.45i)7-s − 0.172·8-s + 0.516·10-s + (0.236 − 0.410i)11-s + (0.589 + 1.02i)13-s + (1.14 + 1.98i)14-s + (0.554 − 0.961i)16-s + 1.01·17-s − 0.313·19-s + (−0.165 + 0.285i)20-s + (0.324 + 0.561i)22-s + (−0.663 − 1.14i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.12694 + 0.0974781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12694 + 0.0974781i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (7.74 - 13.4i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (52.7 + 91.3i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-761. + 1.31e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.04e3 + 1.81e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-4.66e3 - 8.08e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 2.06e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.36e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (3.86e4 + 6.70e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-8.14e4 + 1.41e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (9.82e3 + 1.70e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-6.85e4 - 1.18e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (6.13e3 - 1.06e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-2.70e5 + 4.67e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.76e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.94e5 - 3.36e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.32e6 - 2.29e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (3.00e5 + 5.20e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.75e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.30e6 + 2.25e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.92e5 + 3.34e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 - 9.20e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.90e6 - 3.30e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−16.33627268665730176059551058526, −14.65468171398489985350315036664, −13.82812441952721758223894945678, −11.85023138032626477884381550791, −10.30777968081480728015552600332, −8.638828548751000322231841825947, −7.67633754116173803973516240268, −6.34812396048042852878458173878, −4.30586777205236860918703343570, −0.805578594831086751356085132335,
1.52780775808144603467658367125, 3.09877728602284930825096950188, 5.62836816161466778385114863285, 8.072501033971667678473384008190, 9.258504238597240076438667487964, 10.65313214652416494116932321207, 11.71626186424004766306913099081, 12.59034674901103248477932909634, 14.64640807964564778275325855639, 15.57130040260903115482462083217
