| L(s) = 1 | − 2·4-s − 5-s − 4·7-s − 2·11-s + 4·13-s + 4·16-s + 6·17-s + 2·20-s − 5·23-s + 25-s + 8·28-s + 5·29-s − 2·31-s + 4·35-s − 2·37-s − 5·41-s + 2·43-s + 4·44-s + 8·47-s + 9·49-s − 8·52-s − 8·53-s + 2·55-s + 9·59-s + 14·61-s − 8·64-s − 4·65-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 1.51·7-s − 0.603·11-s + 1.10·13-s + 16-s + 1.45·17-s + 0.447·20-s − 1.04·23-s + 1/5·25-s + 1.51·28-s + 0.928·29-s − 0.359·31-s + 0.676·35-s − 0.328·37-s − 0.780·41-s + 0.304·43-s + 0.603·44-s + 1.16·47-s + 9/7·49-s − 1.10·52-s − 1.09·53-s + 0.269·55-s + 1.17·59-s + 1.79·61-s − 64-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.215814308042038816251041128104, −7.46919798876677704904774614467, −6.52237551578787055496477374707, −5.80162394695748290742648774159, −5.18459552064548713918358803217, −3.96159287573029017397572295383, −3.61877568478164620939900488021, −2.78066831964469692275919502966, −1.07675582631089953203114242220, 0,
1.07675582631089953203114242220, 2.78066831964469692275919502966, 3.61877568478164620939900488021, 3.96159287573029017397572295383, 5.18459552064548713918358803217, 5.80162394695748290742648774159, 6.52237551578787055496477374707, 7.46919798876677704904774614467, 8.215814308042038816251041128104
