| L(s) = 1 | + 4·7-s + 4·9-s + 4·23-s − 2·25-s + 8·31-s − 8·41-s − 20·47-s − 12·49-s + 16·63-s − 8·71-s + 16·73-s + 32·79-s + 6·81-s + 8·89-s − 16·97-s − 28·103-s − 24·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 4/3·9-s + 0.834·23-s − 2/5·25-s + 1.43·31-s − 1.24·41-s − 2.91·47-s − 1.71·49-s + 2.01·63-s − 0.949·71-s + 1.87·73-s + 3.60·79-s + 2/3·81-s + 0.847·89-s − 1.62·97-s − 2.75·103-s − 2.25·113-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.26·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818790529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818790529\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $D_{4}$ | \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.410217297651719987564535040589, −9.387370989585894219543836341344, −8.976072389013114517911942541056, −8.431501346196820914516609083895, −8.288051536299278320481287814306, −8.163922623914325870223683387448, −7.75994293805868588706149594218, −7.72398672422804204707184318497, −7.37910142044265702757719372060, −6.66801448540019197078717706831, −6.62208857896194916224533283059, −6.52102605241615628405615221240, −6.34632751921245849871945507548, −5.38663634296394194511571409250, −5.23635057618917991931870911009, −5.09947870171537809908534329092, −4.82806032479817054719576933332, −4.31977732382427068268194869951, −4.15431509374419544021751980963, −3.69548444200860688675306498822, −3.06159837411850381180474499274, −2.94655939220162838838456849961, −1.93937176132492931752819808343, −1.77346204202998070979589408106, −1.19825709651491645034593071036,
1.19825709651491645034593071036, 1.77346204202998070979589408106, 1.93937176132492931752819808343, 2.94655939220162838838456849961, 3.06159837411850381180474499274, 3.69548444200860688675306498822, 4.15431509374419544021751980963, 4.31977732382427068268194869951, 4.82806032479817054719576933332, 5.09947870171537809908534329092, 5.23635057618917991931870911009, 5.38663634296394194511571409250, 6.34632751921245849871945507548, 6.52102605241615628405615221240, 6.62208857896194916224533283059, 6.66801448540019197078717706831, 7.37910142044265702757719372060, 7.72398672422804204707184318497, 7.75994293805868588706149594218, 8.163922623914325870223683387448, 8.288051536299278320481287814306, 8.431501346196820914516609083895, 8.976072389013114517911942541056, 9.387370989585894219543836341344, 9.410217297651719987564535040589
