L(s) = 1 | − 1.03·3-s + 2.53·5-s − 0.333·7-s − 1.92·9-s − 4.00·11-s − 1.11·13-s − 2.62·15-s + 7.29·17-s − 5.63·19-s + 0.345·21-s − 4.25·23-s + 1.41·25-s + 5.10·27-s − 1.89·29-s + 9.43·31-s + 4.15·33-s − 0.844·35-s − 0.479·37-s + 1.15·39-s − 2.92·41-s − 2.67·43-s − 4.87·45-s − 5.38·47-s − 6.88·49-s − 7.56·51-s + 11.0·53-s − 10.1·55-s + ⋯ |
L(s) = 1 | − 0.598·3-s + 1.13·5-s − 0.126·7-s − 0.641·9-s − 1.20·11-s − 0.309·13-s − 0.677·15-s + 1.76·17-s − 1.29·19-s + 0.0754·21-s − 0.887·23-s + 0.282·25-s + 0.982·27-s − 0.352·29-s + 1.69·31-s + 0.723·33-s − 0.142·35-s − 0.0787·37-s + 0.185·39-s − 0.457·41-s − 0.407·43-s − 0.726·45-s − 0.784·47-s − 0.984·49-s − 1.05·51-s + 1.52·53-s − 1.36·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.382121338\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.382121338\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 1.03T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 + 0.333T + 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 + 5.63T + 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 - 9.43T + 31T^{2} \) |
| 37 | \( 1 + 0.479T + 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 9.57T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 - 0.712T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
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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
−8.215609096857695463400697382734, −7.34581238553858633190406701802, −6.34313324303759459841775147875, −5.95885620917521475195422913666, −5.32198005562719689077854143603, −4.78392452337207338853561615508, −3.53225102057134022198037443517, −2.63536117353098769375856741367, −1.96244202829029085271284263328, −0.61352917675006358825260561254,
0.61352917675006358825260561254, 1.96244202829029085271284263328, 2.63536117353098769375856741367, 3.53225102057134022198037443517, 4.78392452337207338853561615508, 5.32198005562719689077854143603, 5.95885620917521475195422913666, 6.34313324303759459841775147875, 7.34581238553858633190406701802, 8.215609096857695463400697382734