| L(s) = 1 | − 2·2-s + 4-s + 4·7-s + 2·11-s − 8·14-s + 16-s − 8·17-s − 8·19-s − 4·22-s − 8·23-s + 4·28-s + 4·29-s + 2·32-s + 16·34-s − 12·37-s + 16·38-s − 4·41-s + 12·43-s + 2·44-s + 16·46-s − 8·47-s + 6·49-s − 4·53-s − 8·58-s + 8·59-s − 12·61-s − 11·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.51·7-s + 0.603·11-s − 2.13·14-s + 1/4·16-s − 1.94·17-s − 1.83·19-s − 0.852·22-s − 1.66·23-s + 0.755·28-s + 0.742·29-s + 0.353·32-s + 2.74·34-s − 1.97·37-s + 2.59·38-s − 0.624·41-s + 1.82·43-s + 0.301·44-s + 2.35·46-s − 1.16·47-s + 6/7·49-s − 0.549·53-s − 1.05·58-s + 1.04·59-s − 1.53·61-s − 1.37·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.586821501530456464839783061432, −8.513919456537368504398299507875, −8.193105262987933401903597802500, −7.916052298769149406857064929715, −7.14448861378128696754356355404, −7.09710671797099572276068316357, −6.48826185665042504057062220566, −6.04352277637619096256287229548, −5.90661617028775874847721765252, −5.07494275294566641492761933675, −4.61189264607412807046133003673, −4.41518627320243611006829144466, −4.07211395306853007804500071477, −3.44384537625838116142222658420, −2.57677189011361346570474515760, −2.20812933505978642288681380919, −1.64464510029904934324792656499, −1.32696562717415949082235892683, 0, 0,
1.32696562717415949082235892683, 1.64464510029904934324792656499, 2.20812933505978642288681380919, 2.57677189011361346570474515760, 3.44384537625838116142222658420, 4.07211395306853007804500071477, 4.41518627320243611006829144466, 4.61189264607412807046133003673, 5.07494275294566641492761933675, 5.90661617028775874847721765252, 6.04352277637619096256287229548, 6.48826185665042504057062220566, 7.09710671797099572276068316357, 7.14448861378128696754356355404, 7.916052298769149406857064929715, 8.193105262987933401903597802500, 8.513919456537368504398299507875, 8.586821501530456464839783061432
