| L(s) = 1 | − 1.83·3-s − 1.51·5-s − 1.20·7-s + 0.368·9-s + 5.83·11-s − 5.40·13-s + 2.77·15-s + 1.92·17-s − 0.965·19-s + 2.20·21-s + 0.470·23-s − 2.71·25-s + 4.83·27-s − 3.67·29-s − 0.675·31-s − 10.7·33-s + 1.81·35-s + 5.66·37-s + 9.92·39-s − 5.00·41-s − 3.18·43-s − 0.556·45-s + 10.1·47-s − 5.55·49-s − 3.53·51-s + 8.25·53-s − 8.83·55-s + ⋯ |
| L(s) = 1 | − 1.05·3-s − 0.676·5-s − 0.454·7-s + 0.122·9-s + 1.76·11-s − 1.49·13-s + 0.716·15-s + 0.467·17-s − 0.221·19-s + 0.481·21-s + 0.0980·23-s − 0.542·25-s + 0.929·27-s − 0.682·29-s − 0.121·31-s − 1.86·33-s + 0.307·35-s + 0.932·37-s + 1.58·39-s − 0.781·41-s − 0.485·43-s − 0.0829·45-s + 1.48·47-s − 0.793·49-s − 0.495·51-s + 1.13·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 + 1.83T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 0.965T + 19T^{2} \) |
| 23 | \( 1 - 0.470T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 + 0.675T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 8.25T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 6.07T + 97T^{2} \) |
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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
−7.18586213323321288290973603957, −6.77288052630609890074174619254, −6.03683464733426261379696091462, −5.43508546887922907577274119910, −4.60330147344770211707205079107, −3.98344783496175868071095002094, −3.23252112125749564506476881462, −2.13039516686741750360166516951, −0.927803088539878341695584814832, 0,
0.927803088539878341695584814832, 2.13039516686741750360166516951, 3.23252112125749564506476881462, 3.98344783496175868071095002094, 4.60330147344770211707205079107, 5.43508546887922907577274119910, 6.03683464733426261379696091462, 6.77288052630609890074174619254, 7.18586213323321288290973603957
