L(s) = 1 | − 2·5-s + 7-s + 2·11-s + 4·13-s − 3·17-s − 19-s + 2·23-s + 3·25-s − 7·29-s − 9·31-s − 2·35-s + 37-s − 4·41-s − 4·43-s − 18·47-s − 9·49-s − 5·53-s − 4·55-s − 6·59-s + 17·61-s − 8·65-s + 4·67-s − 5·71-s − 4·73-s + 2·77-s + 6·79-s + 2·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.603·11-s + 1.10·13-s − 0.727·17-s − 0.229·19-s + 0.417·23-s + 3/5·25-s − 1.29·29-s − 1.61·31-s − 0.338·35-s + 0.164·37-s − 0.624·41-s − 0.609·43-s − 2.62·47-s − 9/7·49-s − 0.686·53-s − 0.539·55-s − 0.781·59-s + 2.17·61-s − 0.992·65-s + 0.488·67-s − 0.593·71-s − 0.468·73-s + 0.227·77-s + 0.675·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 17 T + 156 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 284 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.58457854169057470028841080312, −7.49239849729328195465932937746, −6.85696114648667995847584918322, −6.72633103393073266004091860030, −6.24890522141297399451946130037, −6.18707418678238553273257205881, −5.46872103272161474476497025779, −5.13674001962234622844669767450, −4.88533401280455117831328871727, −4.51536497150318530782591768223, −3.92761848324211068144625572683, −3.73449985364262156422424914916, −3.39057917154115975886734368326, −3.17931771598565961749959093966, −2.31005459100045372567236358959, −2.00734894159940210734420942036, −1.41290922155230762443110682275, −1.17787551093557067709920220950, 0, 0,
1.17787551093557067709920220950, 1.41290922155230762443110682275, 2.00734894159940210734420942036, 2.31005459100045372567236358959, 3.17931771598565961749959093966, 3.39057917154115975886734368326, 3.73449985364262156422424914916, 3.92761848324211068144625572683, 4.51536497150318530782591768223, 4.88533401280455117831328871727, 5.13674001962234622844669767450, 5.46872103272161474476497025779, 6.18707418678238553273257205881, 6.24890522141297399451946130037, 6.72633103393073266004091860030, 6.85696114648667995847584918322, 7.49239849729328195465932937746, 7.58457854169057470028841080312