L(s) = 1 | + 1.49·3-s + 5-s − 3.29·7-s − 0.764·9-s + 2.43·11-s − 6.19·13-s + 1.49·15-s + 4.65·17-s + 4.81·19-s − 4.92·21-s − 2.74·23-s + 25-s − 5.62·27-s − 0.0387·29-s + 4.95·31-s + 3.63·33-s − 3.29·35-s + 0.899·37-s − 9.26·39-s + 6.32·41-s − 6.14·43-s − 0.764·45-s − 3.35·47-s + 3.83·49-s + 6.95·51-s − 5.39·53-s + 2.43·55-s + ⋯ |
L(s) = 1 | + 0.863·3-s + 0.447·5-s − 1.24·7-s − 0.254·9-s + 0.733·11-s − 1.71·13-s + 0.386·15-s + 1.12·17-s + 1.10·19-s − 1.07·21-s − 0.571·23-s + 0.200·25-s − 1.08·27-s − 0.00719·29-s + 0.889·31-s + 0.633·33-s − 0.556·35-s + 0.147·37-s − 1.48·39-s + 0.988·41-s − 0.937·43-s − 0.113·45-s − 0.489·47-s + 0.548·49-s + 0.974·51-s − 0.741·53-s + 0.328·55-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 0.0387T + 29T^{2} \) |
| 31 | \( 1 - 4.95T + 31T^{2} \) |
| 37 | \( 1 - 0.899T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 + 6.14T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.78T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 7.85T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 - 1.00T + 89T^{2} \) |
| 97 | \( 1 + 9.09T + 97T^{2} \) |
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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
−7.61502769055110823519878545701, −6.83568240586164900540741346144, −6.13565281840144181754058655400, −5.48871396857327326759384804034, −4.63445194247561974320683609893, −3.61750470863657075510966849718, −3.00807761585387028722931177486, −2.52925241468497724792894885476, −1.37992304094099461188320787442, 0,
1.37992304094099461188320787442, 2.52925241468497724792894885476, 3.00807761585387028722931177486, 3.61750470863657075510966849718, 4.63445194247561974320683609893, 5.48871396857327326759384804034, 6.13565281840144181754058655400, 6.83568240586164900540741346144, 7.61502769055110823519878545701