| L(s) = 1 | + (−2.72 − 0.0352i)2-s + (−1.51 − 0.848i)3-s + (5.42 + 0.140i)4-s + (0.633 + 0.194i)5-s + (4.08 + 2.36i)6-s + (1.09 − 2.71i)7-s + (−9.33 − 0.362i)8-s + (1.56 + 2.56i)9-s + (−1.71 − 0.551i)10-s + (1.46 + 0.833i)11-s + (−8.07 − 4.81i)12-s + (2.23 + 4.99i)13-s + (−3.06 + 7.37i)14-s + (−0.792 − 0.830i)15-s + (14.5 + 0.754i)16-s + (1.23 + 0.560i)17-s + ⋯ |
| L(s) = 1 | + (−1.92 − 0.0249i)2-s + (−0.871 − 0.489i)3-s + (2.71 + 0.0701i)4-s + (0.283 + 0.0868i)5-s + (1.66 + 0.965i)6-s + (0.412 − 1.02i)7-s + (−3.29 − 0.128i)8-s + (0.520 + 0.854i)9-s + (−0.543 − 0.174i)10-s + (0.442 + 0.251i)11-s + (−2.33 − 1.38i)12-s + (0.619 + 1.38i)13-s + (−0.819 + 1.96i)14-s + (−0.204 − 0.214i)15-s + (3.64 + 0.188i)16-s + (0.299 + 0.135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.419163 + 0.171532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.419163 + 0.171532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.51 + 0.848i)T \) |
| good | 2 | \( 1 + (2.72 + 0.0352i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (-0.633 - 0.194i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 2.71i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (-1.46 - 0.833i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (-2.23 - 4.99i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 0.560i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.81 - 2.64i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (0.783 - 2.38i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (6.01 - 3.84i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.42 - 5.25i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (9.92 + 1.94i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (4.95 - 1.87i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (-5.12 + 2.25i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (2.78 - 10.2i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (-2.74 + 2.90i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-14.3 + 2.43i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (1.85 - 3.41i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (7.07 - 3.66i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (-6.79 - 5.26i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (-2.20 + 0.706i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (2.76 - 14.5i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (0.646 + 3.96i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (-2.95 - 1.20i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (0.516 - 2.24i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−10.39317567527895576445964913297, −9.776579489881594397125033737859, −8.818110080156140048739245606506, −7.86544261593723933398376670119, −7.09489127145807550389215359420, −6.63669608253767578706911813907, −5.58096931679461341402608749180, −3.86554716319718515497524472580, −1.84503108991395614206581321580, −1.25508710268152638417457603099,
0.54821360954445241703244515047, 1.95119456717149328270109391587, 3.38479314213522479926645147943, 5.47009495550934425031977368191, 5.87362940242277138396629273166, 6.93820978672911646530524513888, 7.977124794969491348444908054291, 8.760528017857999230225736305914, 9.441573458982388492688508803240, 10.17961038544753071517722243970
