L(s) = 1 | + 5.01·2-s + 27·3-s − 102.·4-s − 77.5·5-s + 135.·6-s − 216.·7-s − 1.15e3·8-s + 729·9-s − 389.·10-s + 3.37e3·11-s − 2.77e3·12-s + 1.24e4·13-s − 1.08e3·14-s − 2.09e3·15-s + 7.34e3·16-s − 2.63e4·17-s + 3.65e3·18-s + 4.14e4·19-s + 7.97e3·20-s − 5.85e3·21-s + 1.69e4·22-s − 4.21e4·23-s − 3.12e4·24-s − 7.21e4·25-s + 6.24e4·26-s + 1.96e4·27-s + 2.22e4·28-s + ⋯ |
L(s) = 1 | + 0.443·2-s + 0.577·3-s − 0.803·4-s − 0.277·5-s + 0.256·6-s − 0.238·7-s − 0.799·8-s + 0.333·9-s − 0.123·10-s + 0.764·11-s − 0.463·12-s + 1.57·13-s − 0.105·14-s − 0.160·15-s + 0.448·16-s − 1.30·17-s + 0.147·18-s + 1.38·19-s + 0.222·20-s − 0.137·21-s + 0.339·22-s − 0.721·23-s − 0.461·24-s − 0.922·25-s + 0.697·26-s + 0.192·27-s + 0.191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 2 | \( 1 - 5.01T + 128T^{2} \) |
| 5 | \( 1 + 77.5T + 7.81e4T^{2} \) |
| 7 | \( 1 + 216.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.37e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.24e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.51e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.39e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.36e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.67e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 4.98e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.87e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.74e3T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.72e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.51e6T + 8.07e13T^{2} \) |
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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
−11.02445962349741346026681472857, −9.544895332759540678233542540937, −8.969565443816026530718028024762, −7.984614648844896121320654567890, −6.59402124628909793204985325307, −5.42279456598151940850555812179, −3.94219159113592814308228928287, −3.50491298625839772572149055604, −1.60141385593376517187035292828, 0,
1.60141385593376517187035292828, 3.50491298625839772572149055604, 3.94219159113592814308228928287, 5.42279456598151940850555812179, 6.59402124628909793204985325307, 7.984614648844896121320654567890, 8.969565443816026530718028024762, 9.544895332759540678233542540937, 11.02445962349741346026681472857