| L(s) = 1 | + 3.55·2-s + 6.62·3-s + 4.64·4-s − 5·5-s + 23.5·6-s − 2.03·7-s − 11.9·8-s + 16.8·9-s − 17.7·10-s − 11·11-s + 30.7·12-s − 18.7·13-s − 7.22·14-s − 33.1·15-s − 79.5·16-s − 128.·17-s + 59.9·18-s + 19·19-s − 23.2·20-s − 13.4·21-s − 39.1·22-s + 68.3·23-s − 79.0·24-s + 25·25-s − 66.7·26-s − 67.0·27-s − 9.43·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 1.27·3-s + 0.580·4-s − 0.447·5-s + 1.60·6-s − 0.109·7-s − 0.527·8-s + 0.624·9-s − 0.562·10-s − 0.301·11-s + 0.740·12-s − 0.400·13-s − 0.137·14-s − 0.570·15-s − 1.24·16-s − 1.83·17-s + 0.785·18-s + 0.229·19-s − 0.259·20-s − 0.139·21-s − 0.379·22-s + 0.620·23-s − 0.672·24-s + 0.200·25-s − 0.503·26-s − 0.478·27-s − 0.0636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 3.55T + 8T^{2} \) |
| 3 | \( 1 - 6.62T + 27T^{2} \) |
| 7 | \( 1 + 2.03T + 343T^{2} \) |
| 13 | \( 1 + 18.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 68.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 43.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 51.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 35.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 374.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 470.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 682.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 636.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 512.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 575.T + 9.12e5T^{2} \) |
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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
−8.971740527039313866570056663434, −8.408926982024338646786847947868, −7.35102256575714214881340708111, −6.59929335064766863030854981985, −5.40475547119648757773564875861, −4.51347314297103953731858621374, −3.74664355706408905871465856343, −2.91064714714915531561851897104, −2.13786685480764230616142093979, 0,
2.13786685480764230616142093979, 2.91064714714915531561851897104, 3.74664355706408905871465856343, 4.51347314297103953731858621374, 5.40475547119648757773564875861, 6.59929335064766863030854981985, 7.35102256575714214881340708111, 8.408926982024338646786847947868, 8.971740527039313866570056663434
