| L(s) = 1 | + (0.233 − 1.32i)2-s + (0.173 + 0.0632i)4-s + (1.26 − 1.06i)5-s + (−2.26 + 0.824i)7-s + (1.47 − 2.54i)8-s + (−1.11 − 1.92i)10-s + (4.55 + 3.82i)11-s + (−0.560 − 3.17i)13-s + (0.564 + 3.19i)14-s + (−2.75 − 2.31i)16-s + (−1.5 − 2.59i)17-s + (3.31 − 5.74i)19-s + (0.286 − 0.104i)20-s + (6.13 − 5.14i)22-s + (2.76 + 1.00i)23-s + ⋯ |
| L(s) = 1 | + (0.165 − 0.938i)2-s + (0.0868 + 0.0316i)4-s + (0.566 − 0.475i)5-s + (−0.856 + 0.311i)7-s + (0.520 − 0.901i)8-s + (−0.352 − 0.609i)10-s + (1.37 + 1.15i)11-s + (−0.155 − 0.881i)13-s + (0.150 + 0.855i)14-s + (−0.688 − 0.577i)16-s + (−0.363 − 0.630i)17-s + (0.761 − 1.31i)19-s + (0.0641 − 0.0233i)20-s + (1.30 − 1.09i)22-s + (0.576 + 0.209i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39908 - 1.48294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39908 - 1.48294i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.233 + 1.32i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 1.06i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.26 - 0.824i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-4.55 - 3.82i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.560 + 3.17i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 + 5.74i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.76 - 1.00i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.224 - 1.27i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.553 - 0.201i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.0209 + 0.0362i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.851 + 4.82i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (3.97 + 3.33i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.51 + 1.27i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + (5.62 - 4.72i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (10.3 - 3.77i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.322 - 1.82i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.75 - 4.77i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.77 - 4.81i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.656 - 3.72i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.692 + 3.92i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.07 - 7.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.199 - 0.167i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.10145563794250809068060625575, −9.433126697201848718506852062004, −8.994894627852061886584731089786, −7.22098309636011430890526056621, −6.85844511780822761006249636687, −5.55015213359492255286211207803, −4.50979878941300395611775285849, −3.34418341427621822412155128061, −2.40932787013470605733790149564, −1.13271226667858826554727503923,
1.63788609229976415449466335929, 3.16180772946964283637857601367, 4.24592421787371157229347102037, 5.70315722221736744252312708234, 6.42981377852962948685426371358, 6.66646595253747315698392740036, 7.85499682585146281811710285423, 8.835232198298730741305678288828, 9.691553288064736191153154113865, 10.55721150739732407003654147536
