| L(s) = 1 | − 0.829·2-s − 1.31·4-s − 5-s + 1.06·7-s + 2.74·8-s + 0.829·10-s − 1.84·11-s + 4.52·13-s − 0.880·14-s + 0.344·16-s + 2.62·17-s − 2.99·19-s + 1.31·20-s + 1.53·22-s + 4.21·23-s + 25-s − 3.75·26-s − 1.39·28-s − 6.40·29-s + 0.777·31-s − 5.78·32-s − 2.17·34-s − 1.06·35-s + 4.66·37-s + 2.48·38-s − 2.74·40-s + 2.10·41-s + ⋯ |
| L(s) = 1 | − 0.586·2-s − 0.655·4-s − 0.447·5-s + 0.401·7-s + 0.971·8-s + 0.262·10-s − 0.556·11-s + 1.25·13-s − 0.235·14-s + 0.0861·16-s + 0.635·17-s − 0.686·19-s + 0.293·20-s + 0.326·22-s + 0.879·23-s + 0.200·25-s − 0.736·26-s − 0.263·28-s − 1.18·29-s + 0.139·31-s − 1.02·32-s − 0.372·34-s − 0.179·35-s + 0.766·37-s + 0.402·38-s − 0.434·40-s + 0.328·41-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
| \(L(1)\) |
\(\approx\) |
\(1.054117934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054117934\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.829T + 2T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 + 2.99T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 + 6.40T + 29T^{2} \) |
| 31 | \( 1 - 0.777T + 31T^{2} \) |
| 37 | \( 1 - 4.66T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.838T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 - 4.99T + 61T^{2} \) |
| 67 | \( 1 + 2.35T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.691892757329523626841896437659, −7.66391563758405162386853927575, −7.47091169991338114570657701603, −6.17251687773553737659184095120, −5.47892265941908095018878229744, −4.56388785924405014167903253588, −3.96413418062050009929971116455, −3.01566197646362397765741518832, −1.65785679128752714387785782798, −0.68245136525738940262532287574,
0.68245136525738940262532287574, 1.65785679128752714387785782798, 3.01566197646362397765741518832, 3.96413418062050009929971116455, 4.56388785924405014167903253588, 5.47892265941908095018878229744, 6.17251687773553737659184095120, 7.47091169991338114570657701603, 7.66391563758405162386853927575, 8.691892757329523626841896437659
