L(s) = 1 | − 2.73·3-s + 4.46·9-s + 2·11-s + 0.732·13-s − 7.46·17-s + 19-s + 1.46·23-s − 3.99·27-s − 3.46·29-s + 4·31-s − 5.46·33-s − 7.66·37-s − 2·39-s + 0.535·41-s + 2.92·43-s + 2.92·47-s − 7·49-s + 20.3·51-s + 11.6·53-s − 2.73·57-s + 1.46·59-s − 6.53·61-s + 4.19·67-s − 4·69-s + 10.9·71-s + 10.3·73-s − 8.39·79-s + ⋯ |
L(s) = 1 | − 1.57·3-s + 1.48·9-s + 0.603·11-s + 0.203·13-s − 1.81·17-s + 0.229·19-s + 0.305·23-s − 0.769·27-s − 0.643·29-s + 0.718·31-s − 0.951·33-s − 1.25·37-s − 0.320·39-s + 0.0836·41-s + 0.446·43-s + 0.427·47-s − 49-s + 2.85·51-s + 1.60·53-s − 0.361·57-s + 0.190·59-s − 0.836·61-s + 0.512·67-s − 0.481·69-s + 1.29·71-s + 1.21·73-s − 0.944·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 7.66T + 37T^{2} \) |
| 41 | \( 1 - 0.535T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.108001808058797947298978454859, −6.90456756639894863252033783107, −6.74075841860756512839471624831, −5.90381953976311534480782780472, −5.18248248265521182432674123906, −4.48327970383893913748327302728, −3.71748707450326020536028965323, −2.30547136571454559142166571311, −1.15754762934150027273408769744, 0,
1.15754762934150027273408769744, 2.30547136571454559142166571311, 3.71748707450326020536028965323, 4.48327970383893913748327302728, 5.18248248265521182432674123906, 5.90381953976311534480782780472, 6.74075841860756512839471624831, 6.90456756639894863252033783107, 8.108001808058797947298978454859