| L(s) = 1 | + 1.81·2-s + 1.28·4-s − 2.81·5-s − 7-s − 1.28·8-s − 5.10·10-s − 3.10·11-s + 13-s − 1.81·14-s − 4.91·16-s + 0.524·17-s + 0.813·19-s − 3.62·20-s − 5.62·22-s − 7.33·23-s + 2.91·25-s + 1.81·26-s − 1.28·28-s − 8.28·29-s + 1.39·31-s − 6.33·32-s + 0.951·34-s + 2.81·35-s − 6.15·37-s + 1.47·38-s + 3.62·40-s + 4.20·41-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.644·4-s − 1.25·5-s − 0.377·7-s − 0.455·8-s − 1.61·10-s − 0.935·11-s + 0.277·13-s − 0.484·14-s − 1.22·16-s + 0.127·17-s + 0.186·19-s − 0.811·20-s − 1.19·22-s − 1.53·23-s + 0.583·25-s + 0.355·26-s − 0.243·28-s − 1.53·29-s + 0.250·31-s − 1.12·32-s + 0.163·34-s + 0.475·35-s − 1.01·37-s + 0.239·38-s + 0.573·40-s + 0.656·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 17 | \( 1 - 0.524T + 17T^{2} \) |
| 19 | \( 1 - 0.813T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−9.942797650879718442544684156605, −8.846617689363323918620099696931, −7.907917651429581848553319472586, −7.18409885839837209466190277603, −6.00029239396226175650572082314, −5.28907668433341721528533328361, −4.10834197149675344612278276159, −3.67573139932574585521627910032, −2.51146758915744126420069995012, 0,
2.51146758915744126420069995012, 3.67573139932574585521627910032, 4.10834197149675344612278276159, 5.28907668433341721528533328361, 6.00029239396226175650572082314, 7.18409885839837209466190277603, 7.907917651429581848553319472586, 8.846617689363323918620099696931, 9.942797650879718442544684156605
