| L(s) = 1 | − 12.4·2-s + 49.0·3-s + 27.2·4-s − 16.1·5-s − 611.·6-s − 704.·7-s + 1.25e3·8-s + 219.·9-s + 200.·10-s + 6.42e3·11-s + 1.33e3·12-s − 9.63e3·13-s + 8.78e3·14-s − 790.·15-s − 1.91e4·16-s − 3.31e4·17-s − 2.73e3·18-s − 4.52e3·19-s − 439.·20-s − 3.45e4·21-s − 8.00e4·22-s − 5.02e4·23-s + 6.15e4·24-s − 7.78e4·25-s + 1.20e5·26-s − 9.65e4·27-s − 1.92e4·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 1.04·3-s + 0.213·4-s − 0.0576·5-s − 1.15·6-s − 0.776·7-s + 0.866·8-s + 0.100·9-s + 0.0634·10-s + 1.45·11-s + 0.223·12-s − 1.21·13-s + 0.855·14-s − 0.0604·15-s − 1.16·16-s − 1.63·17-s − 0.110·18-s − 0.151·19-s − 0.0122·20-s − 0.814·21-s − 1.60·22-s − 0.860·23-s + 0.909·24-s − 0.996·25-s + 1.34·26-s − 0.943·27-s − 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 - 5.06e4T \) |
| good | 2 | \( 1 + 12.4T + 128T^{2} \) |
| 3 | \( 1 - 49.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 16.1T + 7.81e4T^{2} \) |
| 7 | \( 1 + 704.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.42e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.63e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.31e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.52e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.02e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.70e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.01e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 2.09e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.31e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.41e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.61e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.12e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.19e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.31e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.27e7T + 8.07e13T^{2} \) |
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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
−14.26866191349589896867353406624, −13.29111766256933302472760864322, −11.61651817940694195743564621638, −9.826589599612862538828365227520, −9.220053168743827305930607621935, −8.139152209905481150992116746905, −6.74179972440062398007586999380, −4.06587337686153894930756808130, −2.14427341254652508906110495602, 0,
2.14427341254652508906110495602, 4.06587337686153894930756808130, 6.74179972440062398007586999380, 8.139152209905481150992116746905, 9.220053168743827305930607621935, 9.826589599612862538828365227520, 11.61651817940694195743564621638, 13.29111766256933302472760864322, 14.26866191349589896867353406624
