| L(s) = 1 | − 1.98·2-s − 0.150·3-s + 1.93·4-s + 2.87·5-s + 0.297·6-s − 2.68·7-s + 0.124·8-s − 2.97·9-s − 5.71·10-s + 0.368·11-s − 0.290·12-s − 2.43·13-s + 5.32·14-s − 0.432·15-s − 4.12·16-s − 3.34·17-s + 5.90·18-s + 0.377·19-s + 5.57·20-s + 0.403·21-s − 0.730·22-s − 0.915·23-s − 0.0186·24-s + 3.28·25-s + 4.83·26-s + 0.897·27-s − 5.20·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 0.0866·3-s + 0.968·4-s + 1.28·5-s + 0.121·6-s − 1.01·7-s + 0.0438·8-s − 0.992·9-s − 1.80·10-s + 0.110·11-s − 0.0839·12-s − 0.676·13-s + 1.42·14-s − 0.111·15-s − 1.03·16-s − 0.811·17-s + 1.39·18-s + 0.0866·19-s + 1.24·20-s + 0.0879·21-s − 0.155·22-s − 0.190·23-s − 0.00380·24-s + 0.657·25-s + 0.948·26-s + 0.172·27-s − 0.982·28-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 | \( 1 + 1.98T + 2T^{2} \) |
| 3 | \( 1 + 0.150T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 - 0.368T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.377T + 19T^{2} \) |
| 23 | \( 1 + 0.915T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 - 5.88T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 - 5.77T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.641T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−10.07266297130011687568116441542, −9.444952104738855267235277160253, −8.927168111408678973448682955011, −7.903134333279320218140783763118, −6.69363422728067934765609913970, −6.12866143400486695929947131990, −4.89035369286801028895846755251, −2.97552901282983377560399798346, −1.87513224418067494690345311532, 0,
1.87513224418067494690345311532, 2.97552901282983377560399798346, 4.89035369286801028895846755251, 6.12866143400486695929947131990, 6.69363422728067934765609913970, 7.903134333279320218140783763118, 8.927168111408678973448682955011, 9.444952104738855267235277160253, 10.07266297130011687568116441542
