| L(s) = 1 | − 17.8·2-s + 27·3-s + 192.·4-s + 255.·5-s − 483.·6-s − 1.07e3·7-s − 1.15e3·8-s + 729·9-s − 4.57e3·10-s + 4.74e3·11-s + 5.19e3·12-s + 5.72e3·13-s + 1.92e4·14-s + 6.89e3·15-s − 3.99e3·16-s − 1.28e4·17-s − 1.30e4·18-s − 5.21e4·19-s + 4.91e4·20-s − 2.89e4·21-s − 8.48e4·22-s − 9.82e3·23-s − 3.11e4·24-s − 1.29e4·25-s − 1.02e5·26-s + 1.96e4·27-s − 2.06e5·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.913·5-s − 0.913·6-s − 1.18·7-s − 0.796·8-s + 0.333·9-s − 1.44·10-s + 1.07·11-s + 0.867·12-s + 0.722·13-s + 1.87·14-s + 0.527·15-s − 0.243·16-s − 0.634·17-s − 0.527·18-s − 1.74·19-s + 1.37·20-s − 0.682·21-s − 1.69·22-s − 0.168·23-s − 0.459·24-s − 0.165·25-s − 1.14·26-s + 0.192·27-s − 1.77·28-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 | \( 1 + 17.8T + 128T^{2} \) |
| 5 | \( 1 - 255.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.07e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.74e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.72e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.28e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.82e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.96e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.72e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.21e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.38e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.33e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.12e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.46e7T + 8.07e13T^{2} \) |
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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
−10.37403316232408012212469337930, −9.666811965923686830076640152606, −9.024267063745054654415431696807, −8.241724768029918615095910680899, −6.70585947019496546490481513452, −6.28588270310964493559292572626, −3.97840215011889821406717030024, −2.42315543647055390817202467667, −1.44268089018856436986875847428, 0,
1.44268089018856436986875847428, 2.42315543647055390817202467667, 3.97840215011889821406717030024, 6.28588270310964493559292572626, 6.70585947019496546490481513452, 8.241724768029918615095910680899, 9.024267063745054654415431696807, 9.666811965923686830076640152606, 10.37403316232408012212469337930
