| L(s) = 1 | − 47.2·2-s + 243·3-s + 189.·4-s + 4.20e3·5-s − 1.14e4·6-s + 8.23e3·7-s + 8.79e4·8-s + 5.90e4·9-s − 1.98e5·10-s + 2.81e5·11-s + 4.59e4·12-s − 2.86e5·13-s − 3.89e5·14-s + 1.02e6·15-s − 4.54e6·16-s + 5.87e6·17-s − 2.79e6·18-s − 8.24e6·19-s + 7.95e5·20-s + 2.00e6·21-s − 1.33e7·22-s − 1.83e7·23-s + 2.13e7·24-s − 3.11e7·25-s + 1.35e7·26-s + 1.43e7·27-s + 1.55e6·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.577·3-s + 0.0923·4-s + 0.601·5-s − 0.603·6-s + 0.185·7-s + 0.948·8-s + 0.333·9-s − 0.628·10-s + 0.527·11-s + 0.0533·12-s − 0.213·13-s − 0.193·14-s + 0.347·15-s − 1.08·16-s + 1.00·17-s − 0.348·18-s − 0.763·19-s + 0.0555·20-s + 0.106·21-s − 0.551·22-s − 0.594·23-s + 0.547·24-s − 0.637·25-s + 0.223·26-s + 0.192·27-s + 0.0171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 + 47.2T + 2.04e3T^{2} \) |
| 5 | \( 1 - 4.20e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.23e3T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.81e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.86e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.87e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.24e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.83e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.47e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 3.33e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.02e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.14e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.42e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.01e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.59e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.49e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.28e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 5.67e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.15e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.09e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.71e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.36e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.29e11T + 7.15e21T^{2} \) |
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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
−9.874117448557412506893481832994, −9.243575599760181596531279497244, −8.282598649951772188856138280615, −7.51235820406061292979456607595, −6.24000163217656322559241339523, −4.84523159091700507924317119431, −3.62514137547653884198230650210, −2.08454272287988560391839945489, −1.32872290754201231267139350424, 0,
1.32872290754201231267139350424, 2.08454272287988560391839945489, 3.62514137547653884198230650210, 4.84523159091700507924317119431, 6.24000163217656322559241339523, 7.51235820406061292979456607595, 8.282598649951772188856138280615, 9.243575599760181596531279497244, 9.874117448557412506893481832994
