| L(s) = 1 | + 87.0·2-s − 243·3-s + 5.53e3·4-s − 4.10e3·5-s − 2.11e4·6-s − 4.94e4·7-s + 3.03e5·8-s + 5.90e4·9-s − 3.57e5·10-s + 6.02e4·11-s − 1.34e6·12-s + 4.34e5·13-s − 4.30e6·14-s + 9.97e5·15-s + 1.51e7·16-s − 9.38e6·17-s + 5.14e6·18-s + 1.79e7·19-s − 2.27e7·20-s + 1.20e7·21-s + 5.24e6·22-s + 4.13e7·23-s − 7.37e7·24-s − 3.19e7·25-s + 3.77e7·26-s − 1.43e7·27-s − 2.73e8·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.577·3-s + 2.70·4-s − 0.587·5-s − 1.11·6-s − 1.11·7-s + 3.27·8-s + 0.333·9-s − 1.13·10-s + 0.112·11-s − 1.56·12-s + 0.324·13-s − 2.14·14-s + 0.339·15-s + 3.60·16-s − 1.60·17-s + 0.641·18-s + 1.66·19-s − 1.58·20-s + 0.642·21-s + 0.217·22-s + 1.33·23-s − 1.89·24-s − 0.655·25-s + 0.623·26-s − 0.192·27-s − 3.00·28-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 | \( 1 + 4.10e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.94e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 6.02e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 4.34e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.38e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.79e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.13e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.20e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.94e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.60e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.02e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.13e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.49e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.26e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 1.28e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.20e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.83e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.47e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.46e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.54e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.02e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.30e10T + 7.15e21T^{2} \) |
| 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
−10.85793200761148520364185138062, −9.360937599142508517351740511239, −7.39108583626961890492711076834, −6.77511492300153603212075583522, −5.79552427349760825655546237071, −4.89651536209279265273541380585, −3.76387829308479475545669338142, −3.13688422753022098929748091222, −1.67783246404550662873027400291, 0,
1.67783246404550662873027400291, 3.13688422753022098929748091222, 3.76387829308479475545669338142, 4.89651536209279265273541380585, 5.79552427349760825655546237071, 6.77511492300153603212075583522, 7.39108583626961890492711076834, 9.360937599142508517351740511239, 10.85793200761148520364185138062
