| L(s) = 1 | + 9.63·2-s − 27·3-s − 35.2·4-s − 113.·5-s − 260.·6-s + 255.·7-s − 1.57e3·8-s + 729·9-s − 1.09e3·10-s − 1.53e3·11-s + 952.·12-s − 3.40e3·13-s + 2.46e3·14-s + 3.05e3·15-s − 1.06e4·16-s − 3.36e4·17-s + 7.02e3·18-s + 5.58e4·19-s + 3.99e3·20-s − 6.89e3·21-s − 1.47e4·22-s − 3.29e4·23-s + 4.24e4·24-s − 6.52e4·25-s − 3.28e4·26-s − 1.96e4·27-s − 9.00e3·28-s + ⋯ |
| L(s) = 1 | + 0.851·2-s − 0.577·3-s − 0.275·4-s − 0.405·5-s − 0.491·6-s + 0.281·7-s − 1.08·8-s + 0.333·9-s − 0.345·10-s − 0.346·11-s + 0.159·12-s − 0.430·13-s + 0.239·14-s + 0.234·15-s − 0.648·16-s − 1.66·17-s + 0.283·18-s + 1.86·19-s + 0.111·20-s − 0.162·21-s − 0.295·22-s − 0.564·23-s + 0.626·24-s − 0.835·25-s − 0.366·26-s − 0.192·27-s − 0.0775·28-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
| \(L(4)\) |
\(\approx\) |
\(0.8657984923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8657984923\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 9.63T + 128T^{2} \) |
| 5 | \( 1 + 113.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 255.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.53e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.40e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.58e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.29e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.65e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.89e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.20e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.42e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.33e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.70e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.92e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.46e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.34e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.37e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.98e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.99e6T + 8.07e13T^{2} \) |
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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
−9.820697980187075006985504208779, −9.096740425006271269419620549128, −7.897228566910199274401362635466, −7.00708947891107491464794368524, −5.80650606976564936196561450146, −5.13657032142608941317668799781, −4.26927811355944630079083618887, −3.37176376348506086246727482875, −1.98428438995507020921209694725, −0.36080396277871294686930346434,
0.36080396277871294686930346434, 1.98428438995507020921209694725, 3.37176376348506086246727482875, 4.26927811355944630079083618887, 5.13657032142608941317668799781, 5.80650606976564936196561450146, 7.00708947891107491464794368524, 7.897228566910199274401362635466, 9.096740425006271269419620549128, 9.820697980187075006985504208779
