| 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
| \(L(1)\) |
\(\approx\) |
\(0.6961990031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6961990031\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.343T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03475684192351090769091152165, −9.488115242187539971656109339633, −9.022859919991342779309901007511, −7.957456576376682828174837836456, −6.90343064278603609270608316095, −6.36122113288789111001909320414, −5.04755137468847749112526420927, −3.71568342102786692878602136320, −2.42874296961364628202601194987, −0.838237704166559149232616522113,
0.838237704166559149232616522113, 2.42874296961364628202601194987, 3.71568342102786692878602136320, 5.04755137468847749112526420927, 6.36122113288789111001909320414, 6.90343064278603609270608316095, 7.957456576376682828174837836456, 9.022859919991342779309901007511, 9.488115242187539971656109339633, 10.03475684192351090769091152165
