| L(s) = 1 | + 10.8·2-s − 243·3-s − 1.92e3·4-s + 8.23e3·5-s − 2.63e3·6-s + 8.59e4·7-s − 4.32e4·8-s + 5.90e4·9-s + 8.94e4·10-s − 4.83e5·11-s + 4.68e5·12-s + 3.96e5·13-s + 9.33e5·14-s − 2.00e6·15-s + 3.48e6·16-s − 1.10e7·17-s + 6.41e5·18-s + 2.99e6·19-s − 1.58e7·20-s − 2.08e7·21-s − 5.25e6·22-s + 1.83e7·23-s + 1.05e7·24-s + 1.90e7·25-s + 4.30e6·26-s − 1.43e7·27-s − 1.65e8·28-s + ⋯ |
| L(s) = 1 | + 0.240·2-s − 0.577·3-s − 0.942·4-s + 1.17·5-s − 0.138·6-s + 1.93·7-s − 0.466·8-s + 0.333·9-s + 0.282·10-s − 0.905·11-s + 0.544·12-s + 0.296·13-s + 0.464·14-s − 0.680·15-s + 0.830·16-s − 1.88·17-s + 0.0800·18-s + 0.277·19-s − 1.11·20-s − 1.11·21-s − 0.217·22-s + 0.594·23-s + 0.269·24-s + 0.389·25-s + 0.0710·26-s − 0.192·27-s − 1.82·28-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 | \( 1 - 10.8T + 2.04e3T^{2} \) |
| 5 | \( 1 - 8.23e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.59e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.83e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 3.96e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.10e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.99e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.83e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.21e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.01e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 8.11e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.31e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.05e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.23e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.80e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 7.57e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.14e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.16e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 9.76e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 8.26e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.53e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.16e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.72e10T + 7.15e21T^{2} \) |
| 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
−10.38471535370922792982720695226, −9.039739322031934553213124906573, −8.380973946758667144363916164485, −6.99431783760853305975591339549, −5.39254944588155212667691949376, −5.24691067818322886280848547809, −4.14843375502697973338551168746, −2.24623665428015532070385560100, −1.36315185981862696033230773389, 0,
1.36315185981862696033230773389, 2.24623665428015532070385560100, 4.14843375502697973338551168746, 5.24691067818322886280848547809, 5.39254944588155212667691949376, 6.99431783760853305975591339549, 8.380973946758667144363916164485, 9.039739322031934553213124906573, 10.38471535370922792982720695226
