L(s) = 1 | + 1.68·2-s + 0.825·4-s − 1.12·5-s − 3.90·7-s − 1.97·8-s − 1.89·10-s − 1.87·11-s − 0.732·13-s − 6.57·14-s − 4.96·16-s + 1.88·17-s + 2.74·19-s − 0.932·20-s − 3.14·22-s − 5.82·23-s − 3.72·25-s − 1.23·26-s − 3.22·28-s − 5.31·29-s + 1.34·31-s − 4.40·32-s + 3.17·34-s + 4.41·35-s + 3.39·37-s + 4.61·38-s + 2.23·40-s + 1.79·41-s + ⋯ |
L(s) = 1 | + 1.18·2-s + 0.412·4-s − 0.505·5-s − 1.47·7-s − 0.698·8-s − 0.600·10-s − 0.563·11-s − 0.203·13-s − 1.75·14-s − 1.24·16-s + 0.458·17-s + 0.629·19-s − 0.208·20-s − 0.670·22-s − 1.21·23-s − 0.744·25-s − 0.241·26-s − 0.609·28-s − 0.987·29-s + 0.240·31-s − 0.778·32-s + 0.544·34-s + 0.746·35-s + 0.558·37-s + 0.747·38-s + 0.352·40-s + 0.280·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.68T + 2T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−9.839206657547177039656060147527, −9.370643373934861421901235159245, −8.065194869266778125089429040670, −7.16177997306440306943283446998, −6.08290033506018708127006184561, −5.53440045323573199354654852474, −4.24543471662823034700266946181, −3.52951825264974116409177435744, −2.62881899175052538848358130059, 0,
2.62881899175052538848358130059, 3.52951825264974116409177435744, 4.24543471662823034700266946181, 5.53440045323573199354654852474, 6.08290033506018708127006184561, 7.16177997306440306943283446998, 8.065194869266778125089429040670, 9.370643373934861421901235159245, 9.839206657547177039656060147527