| L(s) = 1 | + 2·2-s − 4·3-s + 2·4-s − 8·6-s + 4·8-s + 4·9-s − 8·12-s + 8·16-s + 8·18-s − 16·24-s + 4·27-s − 8·31-s + 8·32-s + 8·36-s − 8·37-s − 8·41-s + 28·43-s − 32·48-s + 20·49-s − 32·53-s + 8·54-s − 16·62-s + 8·64-s + 36·67-s + 8·71-s + 16·72-s − 16·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 4-s − 3.26·6-s + 1.41·8-s + 4/3·9-s − 2.30·12-s + 2·16-s + 1.88·18-s − 3.26·24-s + 0.769·27-s − 1.43·31-s + 1.41·32-s + 4/3·36-s − 1.31·37-s − 1.24·41-s + 4.26·43-s − 4.61·48-s + 20/7·49-s − 4.39·53-s + 1.08·54-s − 2.03·62-s + 64-s + 4.39·67-s + 0.949·71-s + 1.88·72-s − 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.164025938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164025938\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1274 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−9.402829178227994909923340988459, −8.810083775405742961499812971304, −8.498446094250028279735558197869, −8.229700196940593816371086644505, −8.083055732987274125402409044524, −7.41399486844540375373296924656, −7.40147535557778742778825504346, −7.10190879812325523363122786735, −6.96736261390371091038807245530, −6.22491983654313444674864798283, −6.16697233812793915684061513162, −6.00134133140784793940463533836, −5.84121737425258478600724503925, −5.33401878785986674608822658003, −5.03814829139291654720665934547, −5.00665771609641931399157047307, −4.90294016326911297116126944033, −4.17188162089093127446360540948, −3.80172237134294341915081086807, −3.73146730323659705067309818438, −3.31947073686882230909682864305, −2.42869210975387180015372082654, −2.34314027565998869961187992019, −1.50849049900254926252380088984, −0.64452322507292513297461392572,
0.64452322507292513297461392572, 1.50849049900254926252380088984, 2.34314027565998869961187992019, 2.42869210975387180015372082654, 3.31947073686882230909682864305, 3.73146730323659705067309818438, 3.80172237134294341915081086807, 4.17188162089093127446360540948, 4.90294016326911297116126944033, 5.00665771609641931399157047307, 5.03814829139291654720665934547, 5.33401878785986674608822658003, 5.84121737425258478600724503925, 6.00134133140784793940463533836, 6.16697233812793915684061513162, 6.22491983654313444674864798283, 6.96736261390371091038807245530, 7.10190879812325523363122786735, 7.40147535557778742778825504346, 7.41399486844540375373296924656, 8.083055732987274125402409044524, 8.229700196940593816371086644505, 8.498446094250028279735558197869, 8.810083775405742961499812971304, 9.402829178227994909923340988459
