| L(s) = 1 | + 5.73e3·3-s + 2.98e5·5-s − 3.55e6·7-s + 1.84e7·9-s − 1.24e7·11-s + 3.29e8·13-s + 1.70e9·15-s + 4.40e8·17-s + 1.73e9·19-s − 2.03e10·21-s + 2.88e10·23-s + 5.83e10·25-s + 2.36e10·27-s + 4.42e10·29-s − 1.76e11·31-s − 7.14e10·33-s − 1.05e12·35-s + 1.08e11·37-s + 1.88e12·39-s − 4.26e11·41-s − 9.74e11·43-s + 5.51e12·45-s + 4.25e12·47-s + 7.87e12·49-s + 2.52e12·51-s + 2.35e11·53-s − 3.71e12·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 1.70·5-s − 1.63·7-s + 1.28·9-s − 0.192·11-s + 1.45·13-s + 2.58·15-s + 0.260·17-s + 0.446·19-s − 2.46·21-s + 1.76·23-s + 1.91·25-s + 0.436·27-s + 0.476·29-s − 1.15·31-s − 0.291·33-s − 2.78·35-s + 0.188·37-s + 2.20·39-s − 0.341·41-s − 0.546·43-s + 2.19·45-s + 1.22·47-s + 1.65·49-s + 0.393·51-s + 0.0275·53-s − 0.329·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(4.547487831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.547487831\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 5.73e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.98e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 3.55e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 1.24e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.29e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 4.40e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.73e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.88e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 4.42e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.76e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.08e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 4.26e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 9.74e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 4.25e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 2.35e11T + 7.31e25T^{2} \) |
| 59 | \( 1 - 4.91e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 1.04e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 4.74e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 8.38e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.57e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.70e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 8.23e13T + 6.11e28T^{2} \) |
| 89 | \( 1 - 2.21e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 6.89e14T + 6.33e29T^{2} \) |
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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
−13.28323040922817287609266268547, −13.10186667705875834141798162154, −10.42424023632212155284893414742, −9.406835855392783714373741053805, −8.832548608024304545555037005521, −6.91467028761653264678787007899, −5.74502848027610153718619736395, −3.43532217312714933784985807926, −2.65252431470071012217599982671, −1.28221855739760611674964585835,
1.28221855739760611674964585835, 2.65252431470071012217599982671, 3.43532217312714933784985807926, 5.74502848027610153718619736395, 6.91467028761653264678787007899, 8.832548608024304545555037005521, 9.406835855392783714373741053805, 10.42424023632212155284893414742, 13.10186667705875834141798162154, 13.28323040922817287609266268547
