| L(s) = 1 | − 0.824·3-s − 2.00·5-s − 1.90·7-s − 2.31·9-s + 1.84·11-s − 3.14·13-s + 1.65·15-s − 1.88·17-s + 5.17·19-s + 1.57·21-s + 2.13·23-s − 0.993·25-s + 4.38·27-s + 4.63·29-s − 1.29·31-s − 1.52·33-s + 3.81·35-s + 2.19·37-s + 2.59·39-s + 3.50·41-s − 7.69·43-s + 4.64·45-s + 7.52·47-s − 3.37·49-s + 1.55·51-s + 0.239·53-s − 3.69·55-s + ⋯ |
| L(s) = 1 | − 0.476·3-s − 0.895·5-s − 0.719·7-s − 0.773·9-s + 0.557·11-s − 0.873·13-s + 0.426·15-s − 0.458·17-s + 1.18·19-s + 0.342·21-s + 0.445·23-s − 0.198·25-s + 0.844·27-s + 0.861·29-s − 0.231·31-s − 0.265·33-s + 0.644·35-s + 0.360·37-s + 0.415·39-s + 0.547·41-s − 1.17·43-s + 0.692·45-s + 1.09·47-s − 0.481·49-s + 0.218·51-s + 0.0328·53-s − 0.498·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.824T + 3T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 - 3.50T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 - 0.239T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 1.59T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 + 8.52T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 8.40T + 97T^{2} \) |
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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
−7.42400295526389782968440836818, −6.78211389430742752389530468781, −6.19409943915091472643143629142, −5.33918003658862873331354808058, −4.74910547357290699112623712151, −3.83739799100530413311891053851, −3.17992914655482888517166796661, −2.40653942026892360040901607019, −0.927087477922664081782185803994, 0,
0.927087477922664081782185803994, 2.40653942026892360040901607019, 3.17992914655482888517166796661, 3.83739799100530413311891053851, 4.74910547357290699112623712151, 5.33918003658862873331354808058, 6.19409943915091472643143629142, 6.78211389430742752389530468781, 7.42400295526389782968440836818
