| L(s) = 1 | + 3.18·3-s − 3.11·5-s − 7-s + 7.13·9-s − 1.84·11-s − 5.61·13-s − 9.90·15-s + 17-s + 4.50·19-s − 3.18·21-s + 6.03·23-s + 4.68·25-s + 13.1·27-s + 7.37·29-s − 0.507·31-s − 5.88·33-s + 3.11·35-s + 10.3·37-s − 17.8·39-s + 0.176·41-s + 2.57·43-s − 22.2·45-s − 9.01·47-s + 49-s + 3.18·51-s + 9.46·53-s + 5.75·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 1.39·5-s − 0.377·7-s + 2.37·9-s − 0.557·11-s − 1.55·13-s − 2.55·15-s + 0.242·17-s + 1.03·19-s − 0.694·21-s + 1.25·23-s + 0.936·25-s + 2.53·27-s + 1.36·29-s − 0.0911·31-s − 1.02·33-s + 0.526·35-s + 1.70·37-s − 2.86·39-s + 0.0275·41-s + 0.393·43-s − 3.31·45-s − 1.31·47-s + 0.142·49-s + 0.445·51-s + 1.30·53-s + 0.776·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.694797398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.694797398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 - 6.03T + 23T^{2} \) |
| 29 | \( 1 - 7.37T + 29T^{2} \) |
| 31 | \( 1 + 0.507T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 0.176T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 + 9.01T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 - 9.30T + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 - 9.87T + 71T^{2} \) |
| 73 | \( 1 + 1.50T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.345086699883556804074953653055, −7.80218912079351183992092328974, −7.35114541379517319049846313519, −6.78929108141975026099633326340, −5.15497554059460812904898814541, −4.47832681428991035758003179012, −3.66583225631843096271255268410, −2.91882570141122372032465381353, −2.49162512720235068673032840135, −0.869016709221692137526318871034,
0.869016709221692137526318871034, 2.49162512720235068673032840135, 2.91882570141122372032465381353, 3.66583225631843096271255268410, 4.47832681428991035758003179012, 5.15497554059460812904898814541, 6.78929108141975026099633326340, 7.35114541379517319049846313519, 7.80218912079351183992092328974, 8.345086699883556804074953653055
