| L(s) = 1 | − 4·2-s + 8·4-s − 4·5-s + 4·7-s − 12·8-s + 16·10-s + 16·11-s − 32·13-s − 16·14-s + 15·16-s − 40·17-s − 32·20-s − 64·22-s + 56·23-s + 16·25-s + 128·26-s + 32·28-s − 16·31-s − 40·32-s + 160·34-s − 16·35-s + 64·37-s + 48·40-s − 56·41-s − 8·43-s + 128·44-s − 224·46-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s − 4/5·5-s + 4/7·7-s − 3/2·8-s + 8/5·10-s + 1.45·11-s − 2.46·13-s − 8/7·14-s + 0.937·16-s − 2.35·17-s − 8/5·20-s − 2.90·22-s + 2.43·23-s + 0.639·25-s + 4.92·26-s + 8/7·28-s − 0.516·31-s − 5/4·32-s + 4.70·34-s − 0.457·35-s + 1.72·37-s + 6/5·40-s − 1.36·41-s − 0.186·43-s + 2.90·44-s − 4.86·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1573066915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1573066915\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{4} T^{5} + p^{7} T^{6} + p^{8} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 156 T^{3} + 2942 T^{4} - 156 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 204 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 9120 T^{3} + 148994 T^{4} + 9120 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 15240 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 50904 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 1722 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 64 T + 2048 T^{2} - 58176 T^{3} + 1440962 T^{4} - 58176 p^{2} T^{5} + 2048 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 3342 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 5256 T^{3} - 557566 T^{4} + 5256 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 506496 p^{2} T^{5} + 8192 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 155064 T^{3} + 15333122 T^{4} - 155064 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 200 T + 20000 T^{2} + 1888200 T^{3} + 153742658 T^{4} + 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} + 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 68 T + p^{2} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 65436 p^{2} T^{5} + 2888 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 101328 T^{3} + 79904642 T^{4} + 101328 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 173820 T^{3} + 150551438 T^{4} + 173820 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−14.75812454433060953282430304298, −14.63331949910544798040443994802, −14.30687367756745500554832403620, −13.40415428810468843990665511310, −13.21971398269982858515992813489, −12.80708065831191322667351679521, −12.15455676244450934468808544599, −11.91071577448152424986163173134, −11.57135144001210906439504254971, −11.19155019164708753230762264607, −10.89979022089326742614525957579, −10.24451000268981907618260024464, −10.11337379201305347934855388501, −9.138486862222952065413425180733, −9.082219061602929262321369428557, −9.033008024282958162235044981256, −8.502508879335912284238454206158, −7.77350152759057367301298746720, −7.17061287013729177032608512564, −7.11450047940662188936748839910, −6.73548236126271204075145647014, −5.59617306224645946076413331450, −4.70631905387741320263632042213, −4.23786525700189323868467818503, −2.58030800527451952960517598202,
2.58030800527451952960517598202, 4.23786525700189323868467818503, 4.70631905387741320263632042213, 5.59617306224645946076413331450, 6.73548236126271204075145647014, 7.11450047940662188936748839910, 7.17061287013729177032608512564, 7.77350152759057367301298746720, 8.502508879335912284238454206158, 9.033008024282958162235044981256, 9.082219061602929262321369428557, 9.138486862222952065413425180733, 10.11337379201305347934855388501, 10.24451000268981907618260024464, 10.89979022089326742614525957579, 11.19155019164708753230762264607, 11.57135144001210906439504254971, 11.91071577448152424986163173134, 12.15455676244450934468808544599, 12.80708065831191322667351679521, 13.21971398269982858515992813489, 13.40415428810468843990665511310, 14.30687367756745500554832403620, 14.63331949910544798040443994802, 14.75812454433060953282430304298
