| L(s) = 1 | + 3.88·2-s − 112.·4-s + 26.2·5-s − 644.·7-s − 936.·8-s + 102.·10-s + 5.69e3·11-s − 606.·13-s − 2.50e3·14-s + 1.08e4·16-s + 3.92e4·17-s − 3.40e4·19-s − 2.96e3·20-s + 2.21e4·22-s + 2.12e4·23-s − 7.74e4·25-s − 2.35e3·26-s + 7.27e4·28-s − 2.61e4·29-s − 3.10e5·31-s + 1.61e5·32-s + 1.52e5·34-s − 1.69e4·35-s − 3.30e5·37-s − 1.32e5·38-s − 2.45e4·40-s − 3.18e5·41-s + ⋯ |
| L(s) = 1 | + 0.343·2-s − 0.882·4-s + 0.0939·5-s − 0.710·7-s − 0.646·8-s + 0.0322·10-s + 1.28·11-s − 0.0765·13-s − 0.243·14-s + 0.659·16-s + 1.93·17-s − 1.13·19-s − 0.0828·20-s + 0.442·22-s + 0.364·23-s − 0.991·25-s − 0.0262·26-s + 0.626·28-s − 0.199·29-s − 1.87·31-s + 0.873·32-s + 0.666·34-s − 0.0667·35-s − 1.07·37-s − 0.391·38-s − 0.0607·40-s − 0.721·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.586233688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586233688\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 3.88T + 128T^{2} \) |
| 5 | \( 1 - 26.2T + 7.81e4T^{2} \) |
| 7 | \( 1 + 644.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.69e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 606.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.12e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.61e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.10e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.30e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.67e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.38e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 3.54e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.67e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.39e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.28e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.46e6T + 8.07e13T^{2} \) |
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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
−9.606091442263311729245627950395, −8.998514052043925096769048508342, −8.020339517908670068711843708216, −6.84467731676927867368616381888, −5.92265689705767761637387816054, −5.11578978481638191483752721465, −3.77865373050874433696317131170, −3.46381123105582048686588072497, −1.74659642780681405564165773603, −0.52874266606029263045689493750,
0.52874266606029263045689493750, 1.74659642780681405564165773603, 3.46381123105582048686588072497, 3.77865373050874433696317131170, 5.11578978481638191483752721465, 5.92265689705767761637387816054, 6.84467731676927867368616381888, 8.020339517908670068711843708216, 8.998514052043925096769048508342, 9.606091442263311729245627950395
