L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 3·9-s + 4·10-s − 4·16-s + 6·18-s − 2·19-s − 4·20-s − 7·23-s − 6·25-s + 4·29-s + 8·32-s − 6·36-s + 4·38-s + 10·43-s + 6·45-s + 14·46-s − 8·47-s − 10·49-s + 12·50-s − 6·53-s − 8·58-s − 8·64-s − 14·67-s − 16·71-s − 4·73-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 9-s + 1.26·10-s − 16-s + 1.41·18-s − 0.458·19-s − 0.894·20-s − 1.45·23-s − 6/5·25-s + 0.742·29-s + 1.41·32-s − 36-s + 0.648·38-s + 1.52·43-s + 0.894·45-s + 2.06·46-s − 1.16·47-s − 1.42·49-s + 1.69·50-s − 0.824·53-s − 1.05·58-s − 64-s − 1.71·67-s − 1.89·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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
−10.93258789280890913806509476878, −10.30441496586516194531812282489, −9.878732908970514702821530762431, −9.248540873216968233368807806891, −8.637911197121902682968399668704, −8.243107408854781492365920333522, −7.71141203017232747055351642250, −7.38521952880572566561119236918, −6.26910332953765805561678817235, −6.00219715771373920394908652491, −4.76672620061633582239851058580, −4.11256588402448402034934941925, −3.09711192308674777900480025227, −1.93356732858540601762976651053, 0,
1.93356732858540601762976651053, 3.09711192308674777900480025227, 4.11256588402448402034934941925, 4.76672620061633582239851058580, 6.00219715771373920394908652491, 6.26910332953765805561678817235, 7.38521952880572566561119236918, 7.71141203017232747055351642250, 8.243107408854781492365920333522, 8.637911197121902682968399668704, 9.248540873216968233368807806891, 9.878732908970514702821530762431, 10.30441496586516194531812282489, 10.93258789280890913806509476878