| L(s) = 1 | + 4·4-s + 4·9-s − 2·16-s − 24·25-s + 16·36-s + 16·37-s + 16·43-s − 36·64-s − 48·67-s + 16·79-s + 2·81-s − 96·100-s + 64·109-s + 64·121-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 64·172-s + ⋯ |
| L(s) = 1 | + 2·4-s + 4/3·9-s − 1/2·16-s − 4.79·25-s + 8/3·36-s + 2.63·37-s + 2.43·43-s − 9/2·64-s − 5.86·67-s + 1.80·79-s + 2/9·81-s − 9.59·100-s + 6.13·109-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s + 4.87·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213224534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213224534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - p T^{2} + 7 T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 12 T^{2} + 84 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 32 T^{2} + 490 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 52 T^{2} + 1204 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 44 T^{2} + 1078 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 72 T^{2} + 2282 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 32 T^{2} + 1546 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 116 T^{2} + 5278 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 108 T^{2} + 6180 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 12 T^{2} - 3234 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 172 T^{2} + 12726 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 108 T^{2} + 9366 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 128 T^{2} + 11536 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 200 T^{2} + 19690 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 144 T^{2} + 14784 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 108 T^{2} + 6894 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 180 T^{2} + 23604 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 144 T^{2} + 19200 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−6.05191121758066991199647938355, −6.03161753070966548676520049550, −5.99808385270719998915408510165, −5.73974811057112963634186723202, −5.72756207853992555131417474541, −5.16232461499685501903693061471, −5.02973151913301200869107537277, −4.78357803533632883834007506455, −4.63470563066117385627352932952, −4.43667159888554517559964762730, −4.32006001289655829341129944719, −4.19509150702599622004624061713, −4.14626243564433909347861378325, −3.90433293438228257629142751242, −3.39839616521953365414459853924, −3.32003398482917202876668138702, −3.27577679978401887057427412308, −2.79716946951251384777466508392, −2.67391096067655859329598825529, −2.32962535916397300617368085890, −2.06326810466203894575535880802, −2.02105912366885799408678448951, −1.83420405875756268406925375809, −1.58757437041983092119635009341, −0.77338498070463203181121372422,
0.77338498070463203181121372422, 1.58757437041983092119635009341, 1.83420405875756268406925375809, 2.02105912366885799408678448951, 2.06326810466203894575535880802, 2.32962535916397300617368085890, 2.67391096067655859329598825529, 2.79716946951251384777466508392, 3.27577679978401887057427412308, 3.32003398482917202876668138702, 3.39839616521953365414459853924, 3.90433293438228257629142751242, 4.14626243564433909347861378325, 4.19509150702599622004624061713, 4.32006001289655829341129944719, 4.43667159888554517559964762730, 4.63470563066117385627352932952, 4.78357803533632883834007506455, 5.02973151913301200869107537277, 5.16232461499685501903693061471, 5.72756207853992555131417474541, 5.73974811057112963634186723202, 5.99808385270719998915408510165, 6.03161753070966548676520049550, 6.05191121758066991199647938355
Plot not available for L-functions of degree greater than 10.
