| L(s) = 1 | + 11.1·2-s − 140.·3-s − 388.·4-s − 992.·5-s − 1.56e3·6-s + 1.05e4·7-s − 1.00e4·8-s + 70.0·9-s − 1.10e4·10-s + 1.31e4·11-s + 5.45e4·12-s + 1.17e5·14-s + 1.39e5·15-s + 8.73e4·16-s − 5.21e5·17-s + 779.·18-s − 9.83e5·19-s + 3.85e5·20-s − 1.48e6·21-s + 1.46e5·22-s − 7.72e5·23-s + 1.40e6·24-s − 9.68e5·25-s + 2.75e6·27-s − 4.10e6·28-s − 5.37e5·29-s + 1.55e6·30-s + ⋯ |
| L(s) = 1 | + 0.491·2-s − 1.00·3-s − 0.758·4-s − 0.709·5-s − 0.492·6-s + 1.66·7-s − 0.864·8-s + 0.00356·9-s − 0.349·10-s + 0.270·11-s + 0.759·12-s + 0.817·14-s + 0.711·15-s + 0.333·16-s − 1.51·17-s + 0.00175·18-s − 1.73·19-s + 0.538·20-s − 1.66·21-s + 0.133·22-s − 0.575·23-s + 0.866·24-s − 0.496·25-s + 0.998·27-s − 1.26·28-s − 0.141·29-s + 0.349·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6083795249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6083795249\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 11.1T + 512T^{2} \) |
| 3 | \( 1 + 140.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 992.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.31e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.21e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.83e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.72e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.37e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.53e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.26e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.44e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.71e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.36e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.66e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.34e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.23e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.40e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.75e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.27e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.06e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.39e8T + 7.60e17T^{2} \) |
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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
−11.37115604193396644070987376538, −10.48653766389034196675013671790, −8.736746095232117077096448150918, −8.243345300112972195476067523827, −6.69132096405336939118286685954, −5.50928252472032588270237629808, −4.62959834361245506111026993171, −3.99405031623920950414103457590, −1.97212604537653278504411367083, −0.37274967091435788208787660415,
0.37274967091435788208787660415, 1.97212604537653278504411367083, 3.99405031623920950414103457590, 4.62959834361245506111026993171, 5.50928252472032588270237629808, 6.69132096405336939118286685954, 8.243345300112972195476067523827, 8.736746095232117077096448150918, 10.48653766389034196675013671790, 11.37115604193396644070987376538
