L(s) = 1 | − 4·7-s − 4·19-s − 2·25-s + 12·29-s − 16·31-s + 4·37-s + 9·49-s + 12·53-s + 12·59-s − 12·83-s + 8·103-s − 28·109-s + 36·113-s + 10·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + 8·175-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 0.917·19-s − 2/5·25-s + 2.22·29-s − 2.87·31-s + 0.657·37-s + 9/7·49-s + 1.64·53-s + 1.56·59-s − 1.31·83-s + 0.788·103-s − 2.68·109-s + 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + 0.604·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.447744340\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.447744340\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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
−8.475164310589725539535958914352, −8.438706800639572008712255992784, −8.019826144864740407884777297903, −7.19223352513606076732536581838, −7.09734154702325248906431418607, −7.04821121628172993232073123278, −6.27874542988212070303393365281, −6.17887122719955139873575357224, −5.67239664290434402623655635383, −5.44434021201073622800100497731, −4.87582852841394568726159052194, −4.29330293750650602807929625854, −4.07564610991572297783634717864, −3.64549085750344638659901625137, −3.03635093879846425250562422419, −2.94825323482482435308327660990, −2.07195135518234775470214142646, −2.00298502874942725079083606883, −0.926629528666825442809814410938, −0.41472991974883482330920704373,
0.41472991974883482330920704373, 0.926629528666825442809814410938, 2.00298502874942725079083606883, 2.07195135518234775470214142646, 2.94825323482482435308327660990, 3.03635093879846425250562422419, 3.64549085750344638659901625137, 4.07564610991572297783634717864, 4.29330293750650602807929625854, 4.87582852841394568726159052194, 5.44434021201073622800100497731, 5.67239664290434402623655635383, 6.17887122719955139873575357224, 6.27874542988212070303393365281, 7.04821121628172993232073123278, 7.09734154702325248906431418607, 7.19223352513606076732536581838, 8.019826144864740407884777297903, 8.438706800639572008712255992784, 8.475164310589725539535958914352