| L(s) = 1 | − 0.381·3-s − 0.618·5-s + 2.85·7-s − 2.85·9-s + 4.61·13-s + 0.236·15-s − 4.61·17-s + 2.85·19-s − 1.09·21-s − 6.47·23-s − 4.61·25-s + 2.23·27-s + 6.38·29-s + 5.85·31-s − 1.76·35-s − 4.85·37-s − 1.76·39-s − 6.38·41-s + 1.76·45-s − 3.61·47-s + 1.14·49-s + 1.76·51-s − 7.09·53-s − 1.09·57-s − 10.0·59-s + 4.61·61-s − 8.14·63-s + ⋯ |
| L(s) = 1 | − 0.220·3-s − 0.276·5-s + 1.07·7-s − 0.951·9-s + 1.28·13-s + 0.0609·15-s − 1.12·17-s + 0.654·19-s − 0.237·21-s − 1.34·23-s − 0.923·25-s + 0.430·27-s + 1.18·29-s + 1.05·31-s − 0.298·35-s − 0.798·37-s − 0.282·39-s − 0.996·41-s + 0.262·45-s − 0.527·47-s + 0.163·49-s + 0.246·51-s − 0.973·53-s − 0.144·57-s − 1.31·59-s + 0.591·61-s − 1.02·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 - 5.85T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 3.61T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 4.61T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 - 8.61T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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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
−7.79539216386094199302181449325, −6.58076219084268386812479722988, −6.24763085755931924077799331098, −5.36442401529314985391387367471, −4.73643223435239162356545661583, −3.97560852899432497302459976466, −3.16940242764768894050474435415, −2.15368105763101421397193769814, −1.29108347921860709782424458771, 0,
1.29108347921860709782424458771, 2.15368105763101421397193769814, 3.16940242764768894050474435415, 3.97560852899432497302459976466, 4.73643223435239162356545661583, 5.36442401529314985391387367471, 6.24763085755931924077799331098, 6.58076219084268386812479722988, 7.79539216386094199302181449325
