| L(s) = 1 | − 4·2-s + 6·4-s − 2·5-s − 4·8-s + 4·9-s + 8·10-s − 4·11-s − 14·13-s + 3·16-s − 12·17-s − 16·18-s − 12·19-s − 12·20-s + 16·22-s + 3·25-s + 56·26-s − 14·29-s − 8·31-s + 48·34-s + 24·36-s − 4·37-s + 48·38-s + 8·40-s − 6·41-s − 4·43-s − 24·44-s − 8·45-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s − 0.894·5-s − 1.41·8-s + 4/3·9-s + 2.52·10-s − 1.20·11-s − 3.88·13-s + 3/4·16-s − 2.91·17-s − 3.77·18-s − 2.75·19-s − 2.68·20-s + 3.41·22-s + 3/5·25-s + 10.9·26-s − 2.59·29-s − 1.43·31-s + 8.23·34-s + 4·36-s − 0.657·37-s + 7.78·38-s + 1.26·40-s − 0.937·41-s − 0.609·43-s − 3.61·44-s − 1.19·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $D_{4}$ | \( ( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 14 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 574 T^{3} + 3276 T^{4} + 574 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 12 T^{2} + 112 T^{3} + 2831 T^{4} + 112 p T^{5} - 12 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 47 T^{2} + 6 T^{3} + 3732 T^{4} + 6 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 72 T^{2} + 8 T^{3} + 5207 T^{4} + 8 p T^{5} - 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 112 T^{3} + 1395 T^{4} - 112 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 14 T + 33 T^{2} + 574 T^{3} + 10892 T^{4} + 574 p T^{5} + 33 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 116 T^{2} + 8967 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 2 T^{2} - 48 T^{3} + 6051 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 10 T - 39 T^{2} + 70 T^{3} + 6692 T^{4} + 70 p T^{5} - 39 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T - 32 T^{2} + 216 T^{3} + 13359 T^{4} + 216 p T^{5} - 32 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16 T + 6 T^{2} - 896 T^{3} + 28259 T^{4} - 896 p T^{5} + 6 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.170301915572458953524296689063, −7.966190924863698471418262415748, −7.61228671327449205766684590908, −7.52028026769307896824607471479, −7.24177386163416309840394217232, −7.21917998714636082607153865935, −7.02804045079666062742447333555, −6.45149239711639396395104514240, −6.44472985891058246765144223269, −6.36846092663247169429493261788, −5.63515935094153939837657519106, −5.49047674192859160471002756438, −5.08562042895818766304510628573, −4.84823248232045327911259167677, −4.53513858379358614263131347168, −4.43367381039975092870509394830, −4.38726002725472160724470477233, −3.95720729443932931772682316162, −3.39633099732029916415493261932, −3.19198563007039513627829142427, −2.63787080760923238035191879374, −2.19592306652031372783635833388, −2.18173533752776634188123322736, −1.95276472145636761828754934479, −1.38472437391578007720764886498, 0, 0, 0, 0,
1.38472437391578007720764886498, 1.95276472145636761828754934479, 2.18173533752776634188123322736, 2.19592306652031372783635833388, 2.63787080760923238035191879374, 3.19198563007039513627829142427, 3.39633099732029916415493261932, 3.95720729443932931772682316162, 4.38726002725472160724470477233, 4.43367381039975092870509394830, 4.53513858379358614263131347168, 4.84823248232045327911259167677, 5.08562042895818766304510628573, 5.49047674192859160471002756438, 5.63515935094153939837657519106, 6.36846092663247169429493261788, 6.44472985891058246765144223269, 6.45149239711639396395104514240, 7.02804045079666062742447333555, 7.21917998714636082607153865935, 7.24177386163416309840394217232, 7.52028026769307896824607471479, 7.61228671327449205766684590908, 7.966190924863698471418262415748, 8.170301915572458953524296689063
