| 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
−10.17961038544753071517722243970, −9.441573458982388492688508803240, −8.760528017857999230225736305914, −7.977124794969491348444908054291, −6.93820978672911646530524513888, −5.87362940242277138396629273166, −5.47009495550934425031977368191, −3.38479314213522479926645147943, −1.95119456717149328270109391587, −0.54821360954445241703244515047,
1.25508710268152638417457603099, 1.84503108991395614206581321580, 3.86554716319718515497524472580, 5.58096931679461341402608749180, 6.63669608253767578706911813907, 7.09489127145807550389215359420, 7.86544261593723933398376670119, 8.818110080156140048739245606506, 9.776579489881594397125033737859, 10.39317567527895576445964913297
