| L(s) = 1 | + (0.0484 + 0.831i)2-s + (1.29 − 0.151i)4-s + (1.17 + 0.278i)5-s + (−0.785 + 1.05i)7-s + (0.478 + 2.71i)8-s + (−0.174 + 0.989i)10-s + (1.49 + 1.58i)11-s + (4.07 − 2.04i)13-s + (−0.915 − 0.601i)14-s + (0.309 − 0.0733i)16-s + (−3.54 + 1.28i)17-s + (−2.50 − 0.911i)19-s + (1.56 + 0.182i)20-s + (−1.24 + 1.31i)22-s + (3.60 + 4.83i)23-s + ⋯ |
| L(s) = 1 | + (0.0342 + 0.588i)2-s + (0.648 − 0.0758i)4-s + (0.524 + 0.124i)5-s + (−0.296 + 0.398i)7-s + (0.169 + 0.958i)8-s + (−0.0551 + 0.312i)10-s + (0.449 + 0.476i)11-s + (1.12 − 0.566i)13-s + (−0.244 − 0.160i)14-s + (0.0773 − 0.0183i)16-s + (−0.859 + 0.312i)17-s + (−0.574 − 0.209i)19-s + (0.349 + 0.0408i)20-s + (−0.264 + 0.280i)22-s + (0.751 + 1.00i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66246 + 1.21678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66246 + 1.21678i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.0484 - 0.831i)T + (-1.98 + 0.232i)T^{2} \) |
| 5 | \( 1 + (-1.17 - 0.278i)T + (4.46 + 2.24i)T^{2} \) |
| 7 | \( 1 + (0.785 - 1.05i)T + (-2.00 - 6.70i)T^{2} \) |
| 11 | \( 1 + (-1.49 - 1.58i)T + (-0.639 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.07 + 2.04i)T + (7.76 - 10.4i)T^{2} \) |
| 17 | \( 1 + (3.54 - 1.28i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (2.50 + 0.911i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-3.60 - 4.83i)T + (-6.59 + 22.0i)T^{2} \) |
| 29 | \( 1 + (3.47 - 2.28i)T + (11.4 - 26.6i)T^{2} \) |
| 31 | \( 1 + (-2.95 + 6.84i)T + (-21.2 - 22.5i)T^{2} \) |
| 37 | \( 1 + (-7.47 - 6.27i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-0.336 + 5.77i)T + (-40.7 - 4.75i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 4.84i)T + (-35.9 - 23.6i)T^{2} \) |
| 47 | \( 1 + (-1.18 - 2.75i)T + (-32.2 + 34.1i)T^{2} \) |
| 53 | \( 1 + (-6.22 + 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.48 - 7.93i)T + (-3.43 - 58.9i)T^{2} \) |
| 61 | \( 1 + (11.7 + 1.36i)T + (59.3 + 14.0i)T^{2} \) |
| 67 | \( 1 + (-0.693 - 0.456i)T + (26.5 + 61.5i)T^{2} \) |
| 71 | \( 1 + (1.25 - 7.14i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.41 + 7.99i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.668 - 11.4i)T + (-78.4 + 9.17i)T^{2} \) |
| 83 | \( 1 + (0.277 + 4.76i)T + (-82.4 + 9.63i)T^{2} \) |
| 89 | \( 1 + (0.578 + 3.28i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.32 + 0.314i)T + (86.6 - 43.5i)T^{2} \) |
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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
−10.67434587609333822334871785119, −9.621115859323509439313182122125, −8.797277827375031809709456805109, −7.87452235623341765877380717503, −6.90575291632292415200510239762, −6.12328993984828428268980526291, −5.61512203216377963147390898028, −4.19911404820732623500896099728, −2.80677194807442541984442104583, −1.70838093834441638066244377772,
1.17388244068171027343467446573, 2.37263309827303770557684661598, 3.54384839704831984277797004358, 4.46114080448796431865190686369, 6.13030646278802678696301025993, 6.45376214125787572262748510932, 7.52658409731625099917605885158, 8.762233020112606559472534628101, 9.408615678264713416297611996331, 10.48008849339267642925860415121
