| L(s) = 1 | + 1.52·2-s − 2.58·3-s + 0.316·4-s + 1.24·5-s − 3.93·6-s − 0.899·7-s − 2.56·8-s + 3.68·9-s + 1.89·10-s + 3.77·11-s − 0.819·12-s − 4.29·13-s − 1.36·14-s − 3.22·15-s − 4.53·16-s − 7.86·17-s + 5.61·18-s − 6.61·19-s + 0.395·20-s + 2.32·21-s + 5.75·22-s + 3.37·23-s + 6.62·24-s − 3.44·25-s − 6.53·26-s − 1.78·27-s − 0.285·28-s + ⋯ |
| L(s) = 1 | + 1.07·2-s − 1.49·3-s + 0.158·4-s + 0.557·5-s − 1.60·6-s − 0.339·7-s − 0.905·8-s + 1.22·9-s + 0.600·10-s + 1.13·11-s − 0.236·12-s − 1.18·13-s − 0.365·14-s − 0.832·15-s − 1.13·16-s − 1.90·17-s + 1.32·18-s − 1.51·19-s + 0.0883·20-s + 0.507·21-s + 1.22·22-s + 0.704·23-s + 1.35·24-s − 0.688·25-s − 1.28·26-s − 0.343·27-s − 0.0538·28-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 | \( 1 - 1.52T + 2T^{2} \) |
| 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 + 0.899T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 + 7.86T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 2.81T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 + 4.86T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 - 2.08T + 47T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 6.55T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 - 9.16T + 97T^{2} \) |
| 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
−10.71825302217784913200482186745, −9.536231123722122125089894829919, −8.890360020130550599050498701657, −6.93605190441621134166151448546, −6.41486283066228522962352600611, −5.68242795770982679480897279594, −4.70879646269567418472658766508, −4.06998371142865556298482755455, −2.27475206779672789357193027850, 0,
2.27475206779672789357193027850, 4.06998371142865556298482755455, 4.70879646269567418472658766508, 5.68242795770982679480897279594, 6.41486283066228522962352600611, 6.93605190441621134166151448546, 8.890360020130550599050498701657, 9.536231123722122125089894829919, 10.71825302217784913200482186745
