| L(s) = 1 | − 2·3-s + 2·9-s + 12·11-s + 8·23-s + 8·25-s − 6·27-s − 24·33-s + 16·37-s + 32·47-s + 16·49-s + 4·59-s − 16·61-s − 16·69-s + 8·71-s − 8·73-s − 16·75-s + 11·81-s + 20·83-s − 16·97-s + 24·99-s − 12·107-s + 16·109-s − 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s + 3.61·11-s + 1.66·23-s + 8/5·25-s − 1.15·27-s − 4.17·33-s + 2.63·37-s + 4.66·47-s + 16/7·49-s + 0.520·59-s − 2.04·61-s − 1.92·69-s + 0.949·71-s − 0.936·73-s − 1.84·75-s + 11/9·81-s + 2.19·83-s − 1.62·97-s + 2.41·99-s − 1.16·107-s + 1.53·109-s − 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.375311004136550745503923058818, −7.81355643235460799238542660951, −7.44684296800419373091218472860, −7.38768480962131930711332998032, −7.32744672378307609304812102459, −6.93886039116066102818692117218, −6.49958337079848265010822460002, −6.46217092559972383772333794662, −6.31772620663705454161691159052, −5.95528125231371632951653869061, −5.79526940032163107620983650896, −5.36493042952581033648915175603, −5.24017022313894892514145531576, −4.78007082550275247062899098846, −4.39857266292059540737170685877, −4.25027968734526827701723628527, −4.06233545444411783821465339472, −3.61708637713807937652164434514, −3.59307890147807467427570262616, −2.86233646484382091941252710760, −2.41150850783916056386223581536, −2.34162863280202814876175064112, −1.21607159322177550095062969573, −1.11252385058558417766764830712, −0.990795964148147302576711378088,
0.990795964148147302576711378088, 1.11252385058558417766764830712, 1.21607159322177550095062969573, 2.34162863280202814876175064112, 2.41150850783916056386223581536, 2.86233646484382091941252710760, 3.59307890147807467427570262616, 3.61708637713807937652164434514, 4.06233545444411783821465339472, 4.25027968734526827701723628527, 4.39857266292059540737170685877, 4.78007082550275247062899098846, 5.24017022313894892514145531576, 5.36493042952581033648915175603, 5.79526940032163107620983650896, 5.95528125231371632951653869061, 6.31772620663705454161691159052, 6.46217092559972383772333794662, 6.49958337079848265010822460002, 6.93886039116066102818692117218, 7.32744672378307609304812102459, 7.38768480962131930711332998032, 7.44684296800419373091218472860, 7.81355643235460799238542660951, 8.375311004136550745503923058818
