L(s) = 1 | − 87.0·2-s − 243·3-s + 5.52e3·4-s − 497.·5-s + 2.11e4·6-s + 1.08e4·7-s − 3.02e5·8-s + 5.90e4·9-s + 4.32e4·10-s + 3.10e5·11-s − 1.34e6·12-s − 1.09e6·13-s − 9.46e5·14-s + 1.20e5·15-s + 1.49e7·16-s + 7.19e6·17-s − 5.13e6·18-s + 6.26e6·19-s − 2.74e6·20-s − 2.64e6·21-s − 2.70e7·22-s + 1.54e7·23-s + 7.34e7·24-s − 4.85e7·25-s + 9.56e7·26-s − 1.43e7·27-s + 6.00e7·28-s + ⋯ |
L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.69·4-s − 0.0712·5-s + 1.10·6-s + 0.244·7-s − 3.26·8-s + 0.333·9-s + 0.136·10-s + 0.582·11-s − 1.55·12-s − 0.820·13-s − 0.470·14-s + 0.0411·15-s + 3.57·16-s + 1.22·17-s − 0.640·18-s + 0.580·19-s − 0.192·20-s − 0.141·21-s − 1.11·22-s + 0.500·23-s + 1.88·24-s − 0.994·25-s + 1.57·26-s − 0.192·27-s + 0.659·28-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 | \( 1 + 87.0T + 2.04e3T^{2} \) |
| 5 | \( 1 + 497.T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.08e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.10e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.09e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.19e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.26e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.54e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 7.10e6T + 1.22e16T^{2} \) |
| 31 | \( 1 - 4.03e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.18e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.85e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.53e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 5.83e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.97e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 4.77e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.18e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.54e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.19e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.35e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.40e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.53e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.41e11T + 7.15e21T^{2} \) |
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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
−9.944499213947863726846447055624, −9.381771055854299665144317685159, −8.137408525010591574234937780786, −7.38877669361174712648824407808, −6.47092140261471750896643654307, −5.28307460190012683297237861513, −3.31197165790883757307915572532, −1.89626068026268920410209236313, −1.01291671806365823166109879874, 0,
1.01291671806365823166109879874, 1.89626068026268920410209236313, 3.31197165790883757307915572532, 5.28307460190012683297237861513, 6.47092140261471750896643654307, 7.38877669361174712648824407808, 8.137408525010591574234937780786, 9.381771055854299665144317685159, 9.944499213947863726846447055624