| L(s) = 1 | − 16·2-s + 233.·3-s + 256·4-s − 356.·5-s − 3.72e3·6-s + 2.40e3·7-s − 4.09e3·8-s + 3.46e4·9-s + 5.70e3·10-s + 7.28e4·11-s + 5.96e4·12-s + 3.93e4·13-s − 3.84e4·14-s − 8.31e4·15-s + 6.55e4·16-s − 5.11e5·17-s − 5.54e5·18-s + 9.00e5·19-s − 9.13e4·20-s + 5.59e5·21-s − 1.16e6·22-s + 3.82e5·23-s − 9.54e5·24-s − 1.82e6·25-s − 6.29e5·26-s + 3.48e6·27-s + 6.14e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s − 0.255·5-s − 1.17·6-s + 0.377·7-s − 0.353·8-s + 1.75·9-s + 0.180·10-s + 1.50·11-s + 0.830·12-s + 0.382·13-s − 0.267·14-s − 0.424·15-s + 0.250·16-s − 1.48·17-s − 1.24·18-s + 1.58·19-s − 0.127·20-s + 0.627·21-s − 1.06·22-s + 0.285·23-s − 0.587·24-s − 0.934·25-s − 0.270·26-s + 1.26·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.082099485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.082099485\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 233.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 356.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 7.28e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.00e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.82e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.93e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.72e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.68e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.45e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.00e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.69e8T + 7.60e17T^{2} \) |
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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
−17.64358468950992880118329828692, −15.86334017545217412238557025563, −14.73286225778781967915854415218, −13.57753063940629120551869376986, −11.53283941259280985314340398088, −9.483992854131210120570944803065, −8.589346429552756975371994559532, −7.16854073133309675418502732780, −3.66154901038404841063752912294, −1.72061557408656683350454753851,
1.72061557408656683350454753851, 3.66154901038404841063752912294, 7.16854073133309675418502732780, 8.589346429552756975371994559532, 9.483992854131210120570944803065, 11.53283941259280985314340398088, 13.57753063940629120551869376986, 14.73286225778781967915854415218, 15.86334017545217412238557025563, 17.64358468950992880118329828692
