| L(s) = 1 | − 2.93·3-s + 5-s − 1.25·7-s + 5.61·9-s − 2.50·11-s + 2.93·13-s − 2.93·15-s + 7.12·17-s − 4.85·19-s + 3.68·21-s + 1.57·23-s + 25-s − 7.68·27-s + 29-s − 4.61·31-s + 7.36·33-s − 1.25·35-s − 9.87·37-s − 8.61·39-s + 0.508·41-s − 1.38·43-s + 5.61·45-s − 1.36·47-s − 5.42·49-s − 20.9·51-s − 2.23·53-s − 2.50·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s + 0.447·5-s − 0.474·7-s + 1.87·9-s − 0.756·11-s + 0.814·13-s − 0.757·15-s + 1.72·17-s − 1.11·19-s + 0.803·21-s + 0.327·23-s + 0.200·25-s − 1.47·27-s + 0.185·29-s − 0.829·31-s + 1.28·33-s − 0.211·35-s − 1.62·37-s − 1.37·39-s + 0.0793·41-s − 0.210·43-s + 0.837·45-s − 0.198·47-s − 0.775·49-s − 2.92·51-s − 0.307·53-s − 0.338·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 1.57T + 23T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 - 0.508T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + 1.36T + 47T^{2} \) |
| 53 | \( 1 + 2.23T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.91T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−7.06931705533546717400160870790, −6.56311255125387578210757106787, −5.91209017111861440109587376267, −5.40638335675322878297960812709, −4.94924088415403238007292198714, −3.90830726588725094383852845775, −3.17684655045528539998396203416, −1.91974995522407848572741257978, −1.01682144007581638035205575672, 0,
1.01682144007581638035205575672, 1.91974995522407848572741257978, 3.17684655045528539998396203416, 3.90830726588725094383852845775, 4.94924088415403238007292198714, 5.40638335675322878297960812709, 5.91209017111861440109587376267, 6.56311255125387578210757106787, 7.06931705533546717400160870790
