| L(s) = 1 | − 13.6·2-s − 25.3·3-s − 325.·4-s + 344.·5-s + 346.·6-s + 2.40e3·7-s + 1.14e4·8-s − 1.90e4·9-s − 4.70e3·10-s + 3.59e4·11-s + 8.24e3·12-s − 2.85e4·13-s − 3.28e4·14-s − 8.73e3·15-s + 1.00e4·16-s − 2.63e5·17-s + 2.60e5·18-s − 5.04e5·19-s − 1.11e5·20-s − 6.09e4·21-s − 4.92e5·22-s − 7.93e4·23-s − 2.90e5·24-s − 1.83e6·25-s + 3.90e5·26-s + 9.82e5·27-s − 7.80e5·28-s + ⋯ |
| L(s) = 1 | − 0.604·2-s − 0.180·3-s − 0.635·4-s + 0.246·5-s + 0.109·6-s + 0.377·7-s + 0.987·8-s − 0.967·9-s − 0.148·10-s + 0.741·11-s + 0.114·12-s − 0.277·13-s − 0.228·14-s − 0.0445·15-s + 0.0382·16-s − 0.764·17-s + 0.584·18-s − 0.888·19-s − 0.156·20-s − 0.0683·21-s − 0.447·22-s − 0.0591·23-s − 0.178·24-s − 0.939·25-s + 0.167·26-s + 0.355·27-s − 0.240·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8905701618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8905701618\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 13.6T + 512T^{2} \) |
| 3 | \( 1 + 25.3T + 1.96e4T^{2} \) |
| 5 | \( 1 - 344.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 3.59e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.63e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.93e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.04e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.46e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.57e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.66e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.44e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.82e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.79e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.35e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.56e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.48e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.87e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.44e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.07e9T + 7.60e17T^{2} \) |
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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
−12.10278797278511906360726356628, −11.06000667755739020357358199553, −9.951864662883008446728507124947, −8.876903370827036304246154272260, −8.150527552058051929745510246474, −6.59201383106892995940949651068, −5.23351844790144707926961496528, −4.00807615693473148297550445843, −2.09880612604627666476616484851, −0.59147610864647509971680316337,
0.59147610864647509971680316337, 2.09880612604627666476616484851, 4.00807615693473148297550445843, 5.23351844790144707926961496528, 6.59201383106892995940949651068, 8.150527552058051929745510246474, 8.876903370827036304246154272260, 9.951864662883008446728507124947, 11.06000667755739020357358199553, 12.10278797278511906360726356628
