| L(s) = 1 | + (0.939 − 0.342i)2-s + (−1.11 − 1.32i)3-s + (0.766 − 0.642i)4-s + (0.439 + 2.49i)5-s + (−1.5 − 0.866i)6-s + (−1.79 − 1.50i)7-s + (0.500 − 0.866i)8-s + (−0.520 + 2.95i)9-s + (1.26 + 2.19i)10-s + (−0.745 + 4.22i)11-s + (−1.70 − 0.300i)12-s + (−0.713 − 0.259i)13-s + (−2.20 − 0.802i)14-s + (2.81 − 3.35i)15-s + (0.173 − 0.984i)16-s + (−2.46 − 4.26i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.642 − 0.766i)3-s + (0.383 − 0.321i)4-s + (0.196 + 1.11i)5-s + (−0.612 − 0.353i)6-s + (−0.679 − 0.570i)7-s + (0.176 − 0.306i)8-s + (−0.173 + 0.984i)9-s + (0.400 + 0.693i)10-s + (−0.224 + 1.27i)11-s + (−0.492 − 0.0868i)12-s + (−0.197 − 0.0719i)13-s + (−0.589 − 0.214i)14-s + (0.727 − 0.867i)15-s + (0.0434 − 0.246i)16-s + (−0.596 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.929566 - 0.278293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.929566 - 0.278293i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.11 + 1.32i)T \) |
| good | 5 | \( 1 + (-0.439 - 2.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.79 + 1.50i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.745 - 4.22i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.713 + 0.259i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.46 + 4.26i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.62 + 6.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.233 - 0.196i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.91 + 1.06i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (6.58 - 5.52i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.60 + 1.67i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.283 - 1.60i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.39 + 1.16i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 0.573T + 53T^{2} \) |
| 59 | \( 1 + (-0.950 - 5.39i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.46 - 7.10i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.0393 + 0.0143i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.10 - 3.64i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.54 + 9.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.92 + 2.52i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (6.41 - 2.33i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.96 - 6.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.570 - 3.23i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−15.11869521607941332312603868635, −13.84702960009547673667086079729, −13.12558336336656766506595488042, −11.91714563610639375103515204358, −10.87797341539091563079830117386, −9.842787852491263120741460873091, −7.11390196016222587747188680258, −6.81778934042816611592621963328, −4.98056117730359741131778298575, −2.70182334989983768221891677530,
3.73362244416975614718113193614, 5.36002748445467669611286149486, 6.11644928847088868153130300080, 8.421337929443840754432665237503, 9.584600478947968357259233567496, 11.05447164511094643243591016677, 12.28217799209658423776321063989, 13.03477811383327586193709992224, 14.47011777915254830561903046478, 15.77621274435088217946463173599
