L(s) = 1 | − 76.9·2-s + 243·3-s + 3.87e3·4-s + 1.56e3·5-s − 1.86e4·6-s + 5.11e4·7-s − 1.40e5·8-s + 5.90e4·9-s − 1.20e5·10-s + 9.13e5·11-s + 9.40e5·12-s + 3.28e4·13-s − 3.93e6·14-s + 3.80e5·15-s + 2.86e6·16-s + 6.52e6·17-s − 4.54e6·18-s + 2.05e7·19-s + 6.06e6·20-s + 1.24e7·21-s − 7.03e7·22-s − 4.51e6·23-s − 3.40e7·24-s − 4.63e7·25-s − 2.52e6·26-s + 1.43e7·27-s + 1.97e8·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.89·4-s + 0.224·5-s − 0.981·6-s + 1.14·7-s − 1.51·8-s + 0.333·9-s − 0.380·10-s + 1.71·11-s + 1.09·12-s + 0.0245·13-s − 1.95·14-s + 0.129·15-s + 0.682·16-s + 1.11·17-s − 0.566·18-s + 1.90·19-s + 0.423·20-s + 0.663·21-s − 2.90·22-s − 0.146·23-s − 0.873·24-s − 0.949·25-s − 0.0417·26-s + 0.192·27-s + 2.17·28-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
\(L(6)\) |
\(\approx\) |
\(2.245674796\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.245674796\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 + 76.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.56e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.11e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 9.13e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 3.28e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.52e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.05e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.51e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.26e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.56e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.39e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.93e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.57e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.59e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.92e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.27e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.16e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.96e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.06e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.72e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.45e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.20e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.48e10T + 7.15e21T^{2} \) |
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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
−10.23145552033671224703091485140, −9.519830077559394591727859308900, −8.757879778046604231983420008303, −7.82945633444600398385330003900, −7.18380030004597866400964745910, −5.79256093601594707492561996915, −4.11490703270431147745012693774, −2.63639162686666859163128104569, −1.29412832606479943872863082017, −1.10542384960922891195550043175,
1.10542384960922891195550043175, 1.29412832606479943872863082017, 2.63639162686666859163128104569, 4.11490703270431147745012693774, 5.79256093601594707492561996915, 7.18380030004597866400964745910, 7.82945633444600398385330003900, 8.757879778046604231983420008303, 9.519830077559394591727859308900, 10.23145552033671224703091485140