| L(s) = 1 | + 3.06·3-s + 5-s + 2.91·7-s + 6.39·9-s + 4.45·11-s + 5.96·13-s + 3.06·15-s − 3.49·17-s − 3.75·19-s + 8.92·21-s − 6.15·23-s + 25-s + 10.4·27-s − 3.40·29-s + 0.910·31-s + 13.6·33-s + 2.91·35-s − 4.70·37-s + 18.2·39-s + 7.05·41-s + 6.11·43-s + 6.39·45-s − 2.41·47-s + 1.47·49-s − 10.7·51-s + 6.82·53-s + 4.45·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s + 0.447·5-s + 1.10·7-s + 2.13·9-s + 1.34·11-s + 1.65·13-s + 0.791·15-s − 0.847·17-s − 0.860·19-s + 1.94·21-s − 1.28·23-s + 0.200·25-s + 2.00·27-s − 0.632·29-s + 0.163·31-s + 2.37·33-s + 0.492·35-s − 0.772·37-s + 2.92·39-s + 1.10·41-s + 0.933·43-s + 0.953·45-s − 0.352·47-s + 0.210·49-s − 1.50·51-s + 0.937·53-s + 0.600·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.089798377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.089798377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 - 5.96T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 - 0.910T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 + 2.41T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 + 7.50T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 1.70T + 73T^{2} \) |
| 79 | \( 1 + 6.72T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.035940030003619003373481928285, −7.37619046118327784338422089853, −6.44155018379958197243120179507, −6.00244336262166489039855819629, −4.69600134141710804841126933740, −3.99187372656324877237372771193, −3.70486689430697537850906157216, −2.52122577996592831649256035175, −1.79485232353002670077767626248, −1.33812153865055286695091227363,
1.33812153865055286695091227363, 1.79485232353002670077767626248, 2.52122577996592831649256035175, 3.70486689430697537850906157216, 3.99187372656324877237372771193, 4.69600134141710804841126933740, 6.00244336262166489039855819629, 6.44155018379958197243120179507, 7.37619046118327784338422089853, 8.035940030003619003373481928285
