L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 6·5-s + 6·6-s − 2·7-s − 3·8-s − 2·9-s + 18·10-s + 11-s − 8·12-s − 4·13-s + 6·14-s + 12·15-s + 3·16-s − 6·17-s + 6·18-s + 19-s − 24·20-s + 4·21-s − 3·22-s − 6·23-s + 6·24-s + 18·25-s + 12·26-s + 10·27-s − 8·28-s + ⋯ |
L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 2.68·5-s + 2.44·6-s − 0.755·7-s − 1.06·8-s − 2/3·9-s + 5.69·10-s + 0.301·11-s − 2.30·12-s − 1.10·13-s + 1.60·14-s + 3.09·15-s + 3/4·16-s − 1.45·17-s + 1.41·18-s + 0.229·19-s − 5.36·20-s + 0.872·21-s − 0.639·22-s − 1.25·23-s + 1.22·24-s + 18/5·25-s + 2.35·26-s + 1.92·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6229 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6229 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 6229 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 125 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T - p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 82 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 4 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 99 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 73 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 109 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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
−17.5141801090, −17.3704096896, −16.8353209340, −16.3896426878, −15.8503021612, −15.6922102646, −15.0434213425, −14.4433970564, −13.6107791985, −12.4776076382, −12.0769496245, −11.9278389573, −11.2528360359, −10.9608073948, −10.2692818627, −9.67199762730, −8.89462827453, −8.46046702832, −8.05478098701, −7.39652759875, −6.84614693268, −6.10804533723, −4.93601810327, −4.09462780235, −3.06355765592, 0, 0,
3.06355765592, 4.09462780235, 4.93601810327, 6.10804533723, 6.84614693268, 7.39652759875, 8.05478098701, 8.46046702832, 8.89462827453, 9.67199762730, 10.2692818627, 10.9608073948, 11.2528360359, 11.9278389573, 12.0769496245, 12.4776076382, 13.6107791985, 14.4433970564, 15.0434213425, 15.6922102646, 15.8503021612, 16.3896426878, 16.8353209340, 17.3704096896, 17.5141801090