| L(s) = 1 | + 2·2-s + 5-s − 3·7-s − 4·8-s + 9-s + 2·10-s − 11-s − 4·13-s − 6·14-s − 4·16-s + 2·17-s + 2·18-s − 2·19-s − 2·22-s + 4·23-s − 8·26-s − 8·29-s − 3·31-s + 4·34-s − 3·35-s − 3·37-s − 4·38-s − 4·40-s − 12·43-s + 45-s + 8·46-s − 2·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.447·5-s − 1.13·7-s − 1.41·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 1.10·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.471·18-s − 0.458·19-s − 0.426·22-s + 0.834·23-s − 1.56·26-s − 1.48·29-s − 0.538·31-s + 0.685·34-s − 0.507·35-s − 0.493·37-s − 0.648·38-s − 0.632·40-s − 1.82·43-s + 0.149·45-s + 1.17·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100341 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100341 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11149 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 17 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T - 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 179 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 122 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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
−14.3064353073, −13.6596231300, −13.2862677791, −13.0116001910, −12.7922281460, −12.4260051142, −11.7741323785, −11.4930085411, −10.6226012437, −10.2211559978, −9.76456440647, −9.47331797115, −8.87111644047, −8.53686281174, −7.56145262737, −7.22734130136, −6.64940560280, −6.04519539870, −5.49051825632, −5.17020785041, −4.59665457139, −3.95624699813, −3.36557443029, −2.89994015688, −1.88909270043, 0,
1.88909270043, 2.89994015688, 3.36557443029, 3.95624699813, 4.59665457139, 5.17020785041, 5.49051825632, 6.04519539870, 6.64940560280, 7.22734130136, 7.56145262737, 8.53686281174, 8.87111644047, 9.47331797115, 9.76456440647, 10.2211559978, 10.6226012437, 11.4930085411, 11.7741323785, 12.4260051142, 12.7922281460, 13.0116001910, 13.2862677791, 13.6596231300, 14.3064353073
