| L(s) = 1 | − 1.04·2-s − 2.71·3-s − 0.908·4-s + 0.714·5-s + 2.83·6-s − 2.03·7-s + 3.03·8-s + 4.38·9-s − 0.745·10-s + 5.43·11-s + 2.46·12-s − 3.96·13-s + 2.12·14-s − 1.94·15-s − 1.35·16-s − 1.30·17-s − 4.58·18-s + 8.24·19-s − 0.648·20-s + 5.52·21-s − 5.67·22-s − 2.44·23-s − 8.25·24-s − 4.49·25-s + 4.14·26-s − 3.76·27-s + 1.84·28-s + ⋯ |
| L(s) = 1 | − 0.738·2-s − 1.56·3-s − 0.454·4-s + 0.319·5-s + 1.15·6-s − 0.768·7-s + 1.07·8-s + 1.46·9-s − 0.235·10-s + 1.63·11-s + 0.712·12-s − 1.10·13-s + 0.567·14-s − 0.501·15-s − 0.339·16-s − 0.317·17-s − 1.08·18-s + 1.89·19-s − 0.145·20-s + 1.20·21-s − 1.21·22-s − 0.510·23-s − 1.68·24-s − 0.898·25-s + 0.812·26-s − 0.725·27-s + 0.349·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 547 | \( 1 + T \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 - 0.714T + 5T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 8.24T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 + 6.02T + 31T^{2} \) |
| 37 | \( 1 + 6.36T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 + 0.932T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 + 7.93T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 9.81T + 73T^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 - 0.276T + 97T^{2} \) |
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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
−10.01737908528731549939561120849, −9.768259699816330321164151765044, −8.933617499905635007068164653641, −7.42814923511835420874764519567, −6.75557946173021316806256849168, −5.74037497357810135460719075010, −4.90646652801495110987904506458, −3.74867835204588607198789804190, −1.42753079084273670382675661256, 0,
1.42753079084273670382675661256, 3.74867835204588607198789804190, 4.90646652801495110987904506458, 5.74037497357810135460719075010, 6.75557946173021316806256849168, 7.42814923511835420874764519567, 8.933617499905635007068164653641, 9.768259699816330321164151765044, 10.01737908528731549939561120849
