L(s) = 1 | + (2.92 + 0.656i)3-s + (0.740 + 2.03i)5-s + (−1.08 − 6.13i)7-s + (8.13 + 3.84i)9-s + (5.10 − 14.0i)11-s + (9.95 + 8.35i)13-s + (0.831 + 6.44i)15-s + (−3.36 − 1.94i)17-s + (−6.39 − 11.0i)19-s + (0.863 − 18.6i)21-s + (35.3 + 6.23i)23-s + (15.5 − 13.0i)25-s + (21.2 + 16.6i)27-s + (7.13 + 8.49i)29-s + (−6.25 + 35.4i)31-s + ⋯ |
L(s) = 1 | + (0.975 + 0.218i)3-s + (0.148 + 0.407i)5-s + (−0.154 − 0.877i)7-s + (0.904 + 0.427i)9-s + (0.464 − 1.27i)11-s + (0.766 + 0.642i)13-s + (0.0554 + 0.429i)15-s + (−0.197 − 0.114i)17-s + (−0.336 − 0.583i)19-s + (0.0411 − 0.889i)21-s + (1.53 + 0.271i)23-s + (0.622 − 0.522i)25-s + (0.788 + 0.614i)27-s + (0.245 + 0.293i)29-s + (−0.201 + 1.14i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0803i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.62492 - 0.105671i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.62492 - 0.105671i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.92 - 0.656i)T \) |
good | 5 | \( 1 + (-0.740 - 2.03i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.08 + 6.13i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-5.10 + 14.0i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-9.95 - 8.35i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (3.36 + 1.94i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (6.39 + 11.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-35.3 - 6.23i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-7.13 - 8.49i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (6.25 - 35.4i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (16.3 - 28.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (14.5 - 17.3i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (58.2 + 21.2i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-3.36 + 0.592i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 79.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (7.36 + 20.2i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-13.7 - 77.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-84.5 - 70.9i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (71.2 + 41.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (3.09 + 5.36i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (49.2 - 41.3i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (97.6 + 116. i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-97.5 + 56.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (52.0 + 18.9i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−10.79053077997373552197678437835, −10.09533987511705942268913337520, −8.804638980374024951412860398004, −8.584346743672030283931386557220, −7.04642767694247771812993362825, −6.61379883536839045515590836802, −4.92357154752235386931493231398, −3.72078739129621843167476394830, −2.98480091304773768539774546854, −1.25207769903529126151262970960,
1.46727207611244296946571112179, 2.65045267928125772793047373223, 3.90499591251431063146879084574, 5.10490807942650109165911658678, 6.36397943882534405643735083607, 7.34574256716310075753224734705, 8.410361145808476805922494431882, 9.062343163298116459208206749455, 9.752490238625548956002357618493, 10.86496822453838944783317753984