| L(s) = 1 | − 10.6·2-s − 27·3-s − 13.5·4-s − 456.·5-s + 288.·6-s − 1.23e3·7-s + 1.51e3·8-s + 729·9-s + 4.88e3·10-s + 2.31e3·11-s + 365.·12-s + 9.27e3·13-s + 1.32e4·14-s + 1.23e4·15-s − 1.44e4·16-s + 2.09e4·17-s − 7.79e3·18-s + 5.00e4·19-s + 6.17e3·20-s + 3.34e4·21-s − 2.47e4·22-s + 5.58e4·23-s − 4.08e4·24-s + 1.30e5·25-s − 9.91e4·26-s − 1.96e4·27-s + 1.67e4·28-s + ⋯ |
| L(s) = 1 | − 0.945·2-s − 0.577·3-s − 0.105·4-s − 1.63·5-s + 0.545·6-s − 1.36·7-s + 1.04·8-s + 0.333·9-s + 1.54·10-s + 0.524·11-s + 0.0609·12-s + 1.17·13-s + 1.29·14-s + 0.942·15-s − 0.883·16-s + 1.03·17-s − 0.315·18-s + 1.67·19-s + 0.172·20-s + 0.788·21-s − 0.496·22-s + 0.957·23-s − 0.603·24-s + 1.66·25-s − 1.10·26-s − 0.192·27-s + 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6156193877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6156193877\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 10.6T + 128T^{2} \) |
| 5 | \( 1 + 456.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.31e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.27e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.00e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.77e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.34e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.99e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.91e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.18e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.60e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.77e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.48e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.79e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.15e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.80e6T + 8.07e13T^{2} \) |
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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
−9.540333124399487238799928480439, −9.244122157046769922452199300708, −7.916942104447135030048591259395, −7.44486101371620591510603964778, −6.42499402998375569680228251192, −5.18000174684982469668600121429, −3.81818053287356277647178098413, −3.41036214455917843059173904779, −1.07732247422349790360786670138, −0.54800539440760830046805337356,
0.54800539440760830046805337356, 1.07732247422349790360786670138, 3.41036214455917843059173904779, 3.81818053287356277647178098413, 5.18000174684982469668600121429, 6.42499402998375569680228251192, 7.44486101371620591510603964778, 7.916942104447135030048591259395, 9.244122157046769922452199300708, 9.540333124399487238799928480439
