| L(s) = 1 | + (0.139 − 0.789i)2-s + (1.27 + 0.464i)4-s + (2.10 − 1.76i)5-s + (−2.23 + 0.812i)7-s + (1.34 − 2.33i)8-s + (−1.10 − 1.90i)10-s + (0.191 + 0.160i)11-s + (0.453 + 2.57i)13-s + (0.330 + 1.87i)14-s + (0.427 + 0.358i)16-s + (−0.146 − 0.254i)17-s + (1.39 − 2.41i)19-s + (3.50 − 1.27i)20-s + (0.153 − 0.128i)22-s + (−6.28 − 2.28i)23-s + ⋯ |
| L(s) = 1 | + (0.0984 − 0.558i)2-s + (0.637 + 0.232i)4-s + (0.942 − 0.790i)5-s + (−0.844 + 0.307i)7-s + (0.475 − 0.823i)8-s + (−0.348 − 0.603i)10-s + (0.0577 + 0.0484i)11-s + (0.125 + 0.713i)13-s + (0.0884 + 0.501i)14-s + (0.106 + 0.0896i)16-s + (−0.0355 − 0.0616i)17-s + (0.319 − 0.553i)19-s + (0.784 − 0.285i)20-s + (0.0327 − 0.0274i)22-s + (−1.31 − 0.477i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47765 - 0.696047i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47765 - 0.696047i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.139 + 0.789i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-2.10 + 1.76i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.23 - 0.812i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.191 - 0.160i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.453 - 2.57i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.146 + 0.254i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.28 + 2.28i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0616 + 0.349i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.59 + 0.945i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.68 - 9.56i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.199 - 0.167i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (10.7 - 3.90i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (4.57 - 3.84i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-11.1 + 4.05i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.314 - 1.78i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.185 + 0.320i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.51 - 4.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.139 + 0.790i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.478 + 2.71i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-5.22 + 9.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.3 + 9.53i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.01247129448258686015537693371, −11.17962106982259479428763850994, −9.848055430511460704617098675618, −9.481469756057982151409819892438, −8.155815559022865234293209298957, −6.71606339424243768808726455361, −5.97425832513267692352354249180, −4.47880633851717940629351402355, −2.97104733726104669677815948823, −1.68486939308645977420167532411,
2.11571442459983756041375546561, 3.47273623236109470478567736395, 5.52105773656339403567076968609, 6.18464927577950935936741429420, 7.02295971519513947989875465609, 8.031169650569591726038698811063, 9.635810539028891026390784335851, 10.27904684560110615727352345146, 11.06201236433851146636502734121, 12.29658307425121395543394531218
