| L(s) = 1 | + (0.367 + 2.08i)2-s + (−2.33 + 0.848i)4-s + (2.05 + 1.72i)5-s + (−0.913 − 0.332i)7-s + (−0.508 − 0.880i)8-s + (−2.83 + 4.91i)10-s + (0.242 − 0.203i)11-s + (−0.262 + 1.49i)13-s + (0.357 − 2.02i)14-s + (−2.15 + 1.80i)16-s + (0.587 − 1.01i)17-s + (−3.11 − 5.38i)19-s + (−6.25 − 2.27i)20-s + (0.513 + 0.431i)22-s + (2.03 − 0.739i)23-s + ⋯ |
| L(s) = 1 | + (0.259 + 1.47i)2-s + (−1.16 + 0.424i)4-s + (0.919 + 0.771i)5-s + (−0.345 − 0.125i)7-s + (−0.179 − 0.311i)8-s + (−0.897 + 1.55i)10-s + (0.0731 − 0.0614i)11-s + (−0.0729 + 0.413i)13-s + (0.0955 − 0.541i)14-s + (−0.538 + 0.451i)16-s + (0.142 − 0.246i)17-s + (−0.713 − 1.23i)19-s + (−1.39 − 0.508i)20-s + (0.109 + 0.0919i)22-s + (0.423 − 0.154i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534954 + 1.41674i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534954 + 1.41674i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.367 - 2.08i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-2.05 - 1.72i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.913 + 0.332i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.242 + 0.203i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.262 - 1.49i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.587 + 1.01i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 + 5.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.03 + 0.739i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.764 - 4.33i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.15 + 2.96i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.23 + 3.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.01 - 5.75i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.28 + 3.59i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (2.32 + 0.846i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (-1.32 - 1.10i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.953 + 0.347i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.148 + 0.843i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.79 - 8.31i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 - 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.94 + 11.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.813 + 4.61i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (7.74 + 13.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.25 - 3.56i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.99502820002503702978494733099, −11.45121889580861494256752105322, −10.41867310353171610929469334475, −9.377663678595439806809320803127, −8.398359117806591979373593171989, −7.02758877101067834934996008052, −6.61142129243883973810201005957, −5.63070711235075331386215997332, −4.43930656052956513692172363484, −2.58863582626795033509905037222,
1.33778468142997423028846564488, 2.64619305693428756062890198981, 4.03885395360661162069839467918, 5.23694700145450967113248960421, 6.38320618196520032975936497747, 8.149102454692176873554246152425, 9.298858184970708323074137206196, 9.983341682837107103211584660339, 10.70988514795102199398488664146, 11.96871116016160670702656541201
