| L(s) = 1 | + 27·3-s + 155.·5-s − 1.74e3·7-s + 729·9-s − 51.6·11-s + 1.04e4·13-s + 4.19e3·15-s + 2.40e4·17-s − 1.14e3·19-s − 4.70e4·21-s − 8.65e4·23-s − 5.40e4·25-s + 1.96e4·27-s − 1.53e5·29-s + 1.32e5·31-s − 1.39e3·33-s − 2.70e5·35-s − 5.04e5·37-s + 2.82e5·39-s + 9.94e4·41-s + 8.43e4·43-s + 1.13e5·45-s − 9.98e5·47-s + 2.21e6·49-s + 6.48e5·51-s − 1.24e6·53-s − 8.02e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.555·5-s − 1.91·7-s + 0.333·9-s − 0.0117·11-s + 1.32·13-s + 0.320·15-s + 1.18·17-s − 0.0384·19-s − 1.10·21-s − 1.48·23-s − 0.691·25-s + 0.192·27-s − 1.16·29-s + 0.800·31-s − 0.00675·33-s − 1.06·35-s − 1.63·37-s + 0.763·39-s + 0.225·41-s + 0.161·43-s + 0.185·45-s − 1.40·47-s + 2.68·49-s + 0.684·51-s − 1.15·53-s − 0.00650·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 - 155.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.74e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 51.6T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.04e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.40e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.14e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.65e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.53e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.32e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.04e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 9.94e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.43e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.24e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.67e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.05e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.16e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.57e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.87e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.96e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.38e5T + 8.07e13T^{2} \) |
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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
−10.34864989668900290612101234244, −9.785893749169103364357865976934, −8.935056142133905938490840088820, −7.73293713523800053261502582224, −6.41056389856563389718325532680, −5.80503667832348388339941018401, −3.80535274720491582520298318201, −3.11244936847674936643152148124, −1.62958698373710952628227809618, 0,
1.62958698373710952628227809618, 3.11244936847674936643152148124, 3.80535274720491582520298318201, 5.80503667832348388339941018401, 6.41056389856563389718325532680, 7.73293713523800053261502582224, 8.935056142133905938490840088820, 9.785893749169103364357865976934, 10.34864989668900290612101234244
