| L(s) = 1 | + 13.8·2-s − 729·3-s − 7.99e3·4-s + 2.43e4·5-s − 1.01e4·6-s − 3.88e4·7-s − 2.24e5·8-s + 5.31e5·9-s + 3.38e5·10-s + 1.36e6·11-s + 5.83e6·12-s − 1.15e7·13-s − 5.39e5·14-s − 1.77e7·15-s + 6.24e7·16-s + 5.54e7·17-s + 7.38e6·18-s − 2.71e8·19-s − 1.94e8·20-s + 2.82e7·21-s + 1.89e7·22-s + 4.08e8·23-s + 1.64e8·24-s − 6.27e8·25-s − 1.61e8·26-s − 3.87e8·27-s + 3.10e8·28-s + ⋯ |
| L(s) = 1 | + 0.153·2-s − 0.577·3-s − 0.976·4-s + 0.697·5-s − 0.0886·6-s − 0.124·7-s − 0.303·8-s + 0.333·9-s + 0.107·10-s + 0.231·11-s + 0.563·12-s − 0.666·13-s − 0.0191·14-s − 0.402·15-s + 0.929·16-s + 0.556·17-s + 0.0511·18-s − 1.32·19-s − 0.680·20-s + 0.0720·21-s + 0.0355·22-s + 0.575·23-s + 0.175·24-s − 0.514·25-s − 0.102·26-s − 0.192·27-s + 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 13.8T + 8.19e3T^{2} \) |
| 5 | \( 1 - 2.43e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.88e4T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.36e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.15e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 5.54e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.71e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 4.08e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.59e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 4.27e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.54e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.20e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.67e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 5.63e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 9.18e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 3.59e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.39e10T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.08e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.32e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.85e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.36e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 6.93e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.40e12T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.825601384116861886420714523137, −9.068736262351196402382103789624, −7.889522394796837834299638930774, −6.54521786345662760162565280473, −5.64128295227301570765520011000, −4.77799490530766575607016778992, −3.77433065642647430875901095861, −2.32111168741963114096624830571, −1.01611019776428907388216151124, 0,
1.01611019776428907388216151124, 2.32111168741963114096624830571, 3.77433065642647430875901095861, 4.77799490530766575607016778992, 5.64128295227301570765520011000, 6.54521786345662760162565280473, 7.889522394796837834299638930774, 9.068736262351196402382103789624, 9.825601384116861886420714523137
