| L(s) = 1 | + 1.63·2-s − 3-s + 0.663·4-s + 1.27·5-s − 1.63·6-s − 7-s − 2.18·8-s + 9-s + 2.08·10-s − 0.411·11-s − 0.663·12-s − 1.47·13-s − 1.63·14-s − 1.27·15-s − 4.88·16-s + 8.04·17-s + 1.63·18-s − 0.275·19-s + 0.846·20-s + 21-s − 0.672·22-s + 6.76·23-s + 2.18·24-s − 3.37·25-s − 2.41·26-s − 27-s − 0.663·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 0.577·3-s + 0.331·4-s + 0.570·5-s − 0.666·6-s − 0.377·7-s − 0.771·8-s + 0.333·9-s + 0.658·10-s − 0.124·11-s − 0.191·12-s − 0.410·13-s − 0.436·14-s − 0.329·15-s − 1.22·16-s + 1.95·17-s + 0.384·18-s − 0.0633·19-s + 0.189·20-s + 0.218·21-s − 0.143·22-s + 1.41·23-s + 0.445·24-s − 0.674·25-s − 0.473·26-s − 0.192·27-s − 0.125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 11 | \( 1 + 0.411T + 11T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 - 8.04T + 17T^{2} \) |
| 19 | \( 1 + 0.275T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 + 7.04T + 29T^{2} \) |
| 31 | \( 1 + 5.15T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 3.97T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 1.09T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 + 0.462T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 8.33T + 97T^{2} \) |
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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
−7.24187872252848509996237677348, −6.55399621785611249590985862357, −5.80881236096030406167088107627, −5.33300906907317295941406636970, −4.98112377572818955257165084802, −3.81549909170091804312904350892, −3.39686442616352524681567713848, −2.46378187100123383460076043466, −1.35400880643932737843553964642, 0,
1.35400880643932737843553964642, 2.46378187100123383460076043466, 3.39686442616352524681567713848, 3.81549909170091804312904350892, 4.98112377572818955257165084802, 5.33300906907317295941406636970, 5.80881236096030406167088107627, 6.55399621785611249590985862357, 7.24187872252848509996237677348
