| L(s) = 1 | + (−0.677 + 1.57i)2-s + (−0.634 − 0.672i)4-s + (−0.0798 − 1.37i)5-s + (−0.301 − 0.0715i)7-s + (−1.72 + 0.629i)8-s + (2.20 + 0.803i)10-s + (2.01 + 1.32i)11-s + (4.58 + 0.536i)13-s + (0.316 − 0.425i)14-s + (0.290 − 4.98i)16-s + (0.161 − 0.135i)17-s + (2.66 + 2.24i)19-s + (−0.871 + 0.923i)20-s + (−3.44 + 2.26i)22-s + (−7.79 + 1.84i)23-s + ⋯ |
| L(s) = 1 | + (−0.478 + 1.11i)2-s + (−0.317 − 0.336i)4-s + (−0.0357 − 0.613i)5-s + (−0.114 − 0.0270i)7-s + (−0.611 + 0.222i)8-s + (0.697 + 0.254i)10-s + (0.607 + 0.399i)11-s + (1.27 + 0.148i)13-s + (0.0846 − 0.113i)14-s + (0.0725 − 1.24i)16-s + (0.0391 − 0.0328i)17-s + (0.612 + 0.513i)19-s + (−0.194 + 0.206i)20-s + (−0.734 + 0.483i)22-s + (−1.62 + 0.385i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.662715 + 0.988460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.662715 + 0.988460i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.677 - 1.57i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.0798 + 1.37i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (0.301 + 0.0715i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (-2.01 - 1.32i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-4.58 - 0.536i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (-0.161 + 0.135i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.66 - 2.24i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (7.79 - 1.84i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (-4.96 - 6.66i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (0.229 - 0.768i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 10.7i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-1.86 - 4.31i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-6.64 + 3.33i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (1.78 + 5.96i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-4.27 - 7.39i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.90 + 3.88i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (6.87 - 7.28i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.749 - 1.00i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (1.68 + 0.611i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (12.6 - 4.61i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.25 - 2.90i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-0.317 + 0.736i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (-12.2 + 4.46i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.387 + 6.65i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.39116528366356041106712433207, −9.506976458352312697877856174790, −8.665617329420328156575083556384, −8.201639495753563614127280043474, −7.16891360659845779093416620132, −6.35718369464548890370446255943, −5.62254230169989757837823297309, −4.43793291840968433373549215200, −3.22543990130897035509291284006, −1.33250860567182586797088414217,
0.835424084178258866600230770825, 2.25750637893758177375259077911, 3.29112813995217786307165094592, 4.14191973753431618987263889761, 5.95662178752923577528280093880, 6.39372556596210942780204744769, 7.72144419431091675174316281357, 8.725239733035058943208959697765, 9.423849208928272071179230667637, 10.30820851907029991422558430960
