| L(s) = 1 | − 2.98·3-s + 3.19·5-s + 7-s + 5.89·9-s − 11-s − 13-s − 9.53·15-s + 1.18·17-s + 0.300·19-s − 2.98·21-s − 7.32·23-s + 5.22·25-s − 8.64·27-s + 3.24·29-s − 6.56·31-s + 2.98·33-s + 3.19·35-s − 4.73·37-s + 2.98·39-s − 4.72·41-s + 8.34·43-s + 18.8·45-s − 3.02·47-s + 49-s − 3.54·51-s + 9.50·53-s − 3.19·55-s + ⋯ |
| L(s) = 1 | − 1.72·3-s + 1.42·5-s + 0.377·7-s + 1.96·9-s − 0.301·11-s − 0.277·13-s − 2.46·15-s + 0.287·17-s + 0.0689·19-s − 0.650·21-s − 1.52·23-s + 1.04·25-s − 1.66·27-s + 0.601·29-s − 1.17·31-s + 0.519·33-s + 0.540·35-s − 0.779·37-s + 0.477·39-s − 0.737·41-s + 1.27·43-s + 2.81·45-s − 0.441·47-s + 0.142·49-s − 0.495·51-s + 1.30·53-s − 0.431·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328087680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328087680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 - 0.300T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 41 | \( 1 + 4.72T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 1.00T + 61T^{2} \) |
| 67 | \( 1 + 7.44T + 67T^{2} \) |
| 71 | \( 1 + 3.06T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.59443473964553056105233373656, −6.91627230615183361385835596575, −6.23239484319941394711903058098, −5.68717310835992034418653297168, −5.31016494862943172496046375068, −4.64634474782400290054159206076, −3.70611252701762111099767429535, −2.29846856472839631272994129202, −1.69096114458614412316609926817, −0.63332007259287476634021488357,
0.63332007259287476634021488357, 1.69096114458614412316609926817, 2.29846856472839631272994129202, 3.70611252701762111099767429535, 4.64634474782400290054159206076, 5.31016494862943172496046375068, 5.68717310835992034418653297168, 6.23239484319941394711903058098, 6.91627230615183361385835596575, 7.59443473964553056105233373656
