| L(s) = 1 | − 3·4-s + 4·5-s − 11-s + 5·16-s − 12·20-s − 16·23-s + 2·25-s − 16·31-s + 12·37-s + 3·44-s − 16·47-s + 2·49-s − 12·53-s − 4·55-s + 8·59-s − 3·64-s − 8·67-s + 20·80-s + 12·89-s + 48·92-s + 4·97-s − 6·100-s + 16·103-s + 12·113-s − 64·115-s + 121-s + 48·124-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.78·5-s − 0.301·11-s + 5/4·16-s − 2.68·20-s − 3.33·23-s + 2/5·25-s − 2.87·31-s + 1.97·37-s + 0.452·44-s − 2.33·47-s + 2/7·49-s − 1.64·53-s − 0.539·55-s + 1.04·59-s − 3/8·64-s − 0.977·67-s + 2.23·80-s + 1.27·89-s + 5.00·92-s + 0.406·97-s − 3/5·100-s + 1.57·103-s + 1.12·113-s − 5.96·115-s + 1/11·121-s + 4.31·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107811 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107811 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.531112831864384789210057232688, −8.938244517857001424659052468572, −8.389268452519734683124359977272, −7.77744707656776841753907290662, −7.60731726518247812897599595849, −6.38866021531112361683475479825, −6.11982556539470572208453322290, −5.64913594526162848677189952595, −5.25909302550772251578024339366, −4.52395871965258729630923756023, −3.98994534926773298366607994601, −3.36662241945177810035682717763, −2.09666333780189433867543562159, −1.82001999803537272093437549398, 0,
1.82001999803537272093437549398, 2.09666333780189433867543562159, 3.36662241945177810035682717763, 3.98994534926773298366607994601, 4.52395871965258729630923756023, 5.25909302550772251578024339366, 5.64913594526162848677189952595, 6.11982556539470572208453322290, 6.38866021531112361683475479825, 7.60731726518247812897599595849, 7.77744707656776841753907290662, 8.389268452519734683124359977272, 8.938244517857001424659052468572, 9.531112831864384789210057232688
