L(s) = 1 | + 2-s + 2.63·3-s + 4-s − 1.08·5-s + 2.63·6-s − 1.77·7-s + 8-s + 3.92·9-s − 1.08·10-s + 3.52·11-s + 2.63·12-s + 2.53·13-s − 1.77·14-s − 2.86·15-s + 16-s + 7.20·17-s + 3.92·18-s + 4.34·19-s − 1.08·20-s − 4.68·21-s + 3.52·22-s − 7.21·23-s + 2.63·24-s − 3.81·25-s + 2.53·26-s + 2.42·27-s − 1.77·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.487·5-s + 1.07·6-s − 0.672·7-s + 0.353·8-s + 1.30·9-s − 0.344·10-s + 1.06·11-s + 0.759·12-s + 0.703·13-s − 0.475·14-s − 0.739·15-s + 0.250·16-s + 1.74·17-s + 0.924·18-s + 0.997·19-s − 0.243·20-s − 1.02·21-s + 0.751·22-s − 1.50·23-s + 0.537·24-s − 0.762·25-s + 0.497·26-s + 0.466·27-s − 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.206107448\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.206107448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 2.63T + 3T^{2} \) |
| 5 | \( 1 + 1.08T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 7.20T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 7.21T + 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 - 2.88T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.195T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 8.71T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 + 7.99T + 71T^{2} \) |
| 73 | \( 1 - 0.279T + 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 - 1.35T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 4.17T + 97T^{2} \) |
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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
−8.233580999889548334653518395410, −7.72787903274334731464251255540, −7.22233289415400274700109557617, −6.09786622603221449813774133699, −5.62215482362503919330936547253, −4.11725918671694514193306022372, −3.73655572772231136228594480555, −3.28149658594844666732782295374, −2.26176994562263849910839111584, −1.20146789906264344009322002323,
1.20146789906264344009322002323, 2.26176994562263849910839111584, 3.28149658594844666732782295374, 3.73655572772231136228594480555, 4.11725918671694514193306022372, 5.62215482362503919330936547253, 6.09786622603221449813774133699, 7.22233289415400274700109557617, 7.72787903274334731464251255540, 8.233580999889548334653518395410