| L(s) = 1 | − 61.5·2-s − 243·3-s + 1.74e3·4-s − 4.11e3·5-s + 1.49e4·6-s + 2.40e4·7-s + 1.87e4·8-s + 5.90e4·9-s + 2.53e5·10-s + 2.36e4·11-s − 4.23e5·12-s − 3.62e5·13-s − 1.48e6·14-s + 1.00e6·15-s − 4.72e6·16-s − 1.56e5·17-s − 3.63e6·18-s + 5.49e6·19-s − 7.18e6·20-s − 5.84e6·21-s − 1.45e6·22-s − 3.77e7·23-s − 4.55e6·24-s − 3.18e7·25-s + 2.23e7·26-s − 1.43e7·27-s + 4.19e7·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 0.577·3-s + 0.851·4-s − 0.589·5-s + 0.785·6-s + 0.541·7-s + 0.202·8-s + 0.333·9-s + 0.802·10-s + 0.0442·11-s − 0.491·12-s − 0.270·13-s − 0.736·14-s + 0.340·15-s − 1.12·16-s − 0.0266·17-s − 0.453·18-s + 0.509·19-s − 0.501·20-s − 0.312·21-s − 0.0602·22-s − 1.22·23-s − 0.116·24-s − 0.652·25-s + 0.368·26-s − 0.192·27-s + 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 243T \) |
| 47 | \( 1 - 2.29e8T \) |
| good | 2 | \( 1 + 61.5T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.11e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.40e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.36e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 3.62e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.56e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 5.49e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.77e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.76e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.06e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.91e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 6.97e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.60e8T + 9.29e17T^{2} \) |
| 53 | \( 1 - 4.19e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.97e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.30e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 8.54e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.59e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.59e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.10e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.79e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.56e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.07e10T + 7.15e21T^{2} \) |
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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
−10.38090385488988745538198489851, −9.600359390803181596976128051220, −8.342608125227649866697370924358, −7.71189484354293483939348923439, −6.63890282940361131323609632014, −5.15020626468091489156502014317, −3.97765747719698570942436647278, −2.10775685854637611068825449194, −0.952040638487077271869195031906, 0,
0.952040638487077271869195031906, 2.10775685854637611068825449194, 3.97765747719698570942436647278, 5.15020626468091489156502014317, 6.63890282940361131323609632014, 7.71189484354293483939348923439, 8.342608125227649866697370924358, 9.600359390803181596976128051220, 10.38090385488988745538198489851
