| L(s) = 1 | + 8·3-s + 32·9-s − 16·19-s + 88·27-s − 32·43-s − 128·57-s + 48·67-s − 24·73-s + 206·81-s + 24·97-s − 16·121-s + 127-s − 256·129-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 512·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4.61·3-s + 32/3·9-s − 3.67·19-s + 16.9·27-s − 4.87·43-s − 16.9·57-s + 5.86·67-s − 2.80·73-s + 22.8·81-s + 2.43·97-s − 1.45·121-s + 0.0887·127-s − 22.5·129-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 + 0.0783·163-s + 0.0773·167-s − 39.1·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230444163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230444163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 142 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 254 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 23 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 1058 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4942 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 3374 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.11443713352006043862102480021, −4.04576556766388938839902927617, −4.01486648521801866076014075052, −3.83539010607705810502087314362, −3.65580733041708669446792084365, −3.54798901044282286655335901112, −3.50029919875936268489268206416, −3.34923932573752241990949956739, −3.29119826995473668273585773132, −3.14589678652872230087628669081, −2.87588818318159540841902644833, −2.61507098647302627806280610524, −2.59723813305697026828723342637, −2.55150938993157031338195770374, −2.45154320679992526817295555630, −2.27765775347073352143906722033, −2.15554462706779686880498111018, −1.86016085896049292601550222150, −1.77400934001150378318835070504, −1.62783638646815293712667854607, −1.62602239377488075194077217682, −1.31468024771070819881859594602, −0.846252547478642396216769495186, −0.70543236259565437707693090412, −0.05989906789557355467684804345,
0.05989906789557355467684804345, 0.70543236259565437707693090412, 0.846252547478642396216769495186, 1.31468024771070819881859594602, 1.62602239377488075194077217682, 1.62783638646815293712667854607, 1.77400934001150378318835070504, 1.86016085896049292601550222150, 2.15554462706779686880498111018, 2.27765775347073352143906722033, 2.45154320679992526817295555630, 2.55150938993157031338195770374, 2.59723813305697026828723342637, 2.61507098647302627806280610524, 2.87588818318159540841902644833, 3.14589678652872230087628669081, 3.29119826995473668273585773132, 3.34923932573752241990949956739, 3.50029919875936268489268206416, 3.54798901044282286655335901112, 3.65580733041708669446792084365, 3.83539010607705810502087314362, 4.01486648521801866076014075052, 4.04576556766388938839902927617, 4.11443713352006043862102480021
Plot not available for L-functions of degree greater than 10.
