| L(s) = 1 | + 8·2-s + 24·4-s + 32·8-s + 48·16-s + 264·25-s + 592·29-s + 1.39e3·37-s + 740·49-s + 2.11e3·50-s + 1.16e3·53-s + 4.73e3·58-s − 1.79e3·64-s + 1.11e4·74-s + 5.92e3·98-s + 6.33e3·100-s + 9.34e3·106-s + 5.04e3·109-s − 1.23e3·113-s + 1.42e4·116-s + 7.20e3·121-s + 127-s − 9.21e3·128-s + 131-s + 137-s + 139-s + 3.34e4·148-s + 149-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s + 1.41·8-s + 3/4·16-s + 2.11·25-s + 3.79·29-s + 6.18·37-s + 2.15·49-s + 5.97·50-s + 3.02·53-s + 10.7·58-s − 7/2·64-s + 17.4·74-s + 6.10·98-s + 6.33·100-s + 8.56·106-s + 4.42·109-s − 1.02·113-s + 11.3·116-s + 5.41·121-s + 0.000698·127-s − 6.36·128-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 18.5·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(110.0231832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(110.0231832\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{2} T + 3 p^{2} T^{2} - p^{5} T^{3} + p^{6} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 740 T^{2} + 5830 p^{2} T^{4} - 740 p^{6} T^{6} + p^{12} T^{8} \) |
| good | 5 | \( ( 1 - 132 T^{2} + 24054 T^{4} - 132 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3604 T^{2} + 6050998 T^{4} - 3604 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 996 T^{2} + 6521622 T^{4} - 996 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 11716 T^{2} + 4268006 p T^{4} - 11716 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 + 1332 p T^{2} + 253777430 T^{4} + 1332 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 24868 T^{2} + 441206182 T^{4} - 24868 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 148 T + 43886 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 96124 T^{2} + 4061075334 T^{4} + 96124 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 348 T + 125310 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 - 217380 T^{2} + 507483702 p T^{4} - 217380 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 - 283668 T^{2} + 32605854006 T^{4} - 283668 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 311612 T^{2} + 44936416582 T^{4} + 311612 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 292 T + 134238 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 + 251196 T^{2} + 86814887798 T^{4} + 251196 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 301092 T^{2} + 33650023638 T^{4} - 301092 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 859380 T^{2} + 348040319190 T^{4} - 859380 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1156068 T^{2} + 580586492646 T^{4} - 1156068 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 1463204 T^{2} + 837552959782 T^{4} - 1463204 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 505476 T^{2} + 120379491654 T^{4} - 505476 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 1019228 T^{2} + 852653167126 T^{4} + 1019228 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 972900 T^{2} + 1038792779622 T^{4} - 972900 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 585348 T^{2} + 538293971334 T^{4} - 585348 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
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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
−4.87647907383505725749952030025, −4.70502370528893507951738886305, −4.60677971329333379998433838659, −4.60100312821744041496756926978, −4.22767280869453675026258740036, −4.13412141401664839718835012810, −4.11532167147719291198302511867, −4.07195637779127095495422876586, −3.80765250014432627294164915546, −3.46679016476891576318566202243, −3.17435395130982011735776171050, −3.14548395090374850241974684114, −3.10921794619692674823756521692, −2.93685253427161721932589385237, −2.65975018653638702929702114511, −2.39875736404682881069508212393, −2.24839016956513153714692151370, −2.16621738089763585129377493763, −2.03946308357283813317215210957, −1.25524335490926524797846955332, −1.24084448970809512818125974727, −0.975610726858598407021592039130, −0.78164187559715192905087243688, −0.62411572148362948113658779062, −0.48990779076566087795949744242,
0.48990779076566087795949744242, 0.62411572148362948113658779062, 0.78164187559715192905087243688, 0.975610726858598407021592039130, 1.24084448970809512818125974727, 1.25524335490926524797846955332, 2.03946308357283813317215210957, 2.16621738089763585129377493763, 2.24839016956513153714692151370, 2.39875736404682881069508212393, 2.65975018653638702929702114511, 2.93685253427161721932589385237, 3.10921794619692674823756521692, 3.14548395090374850241974684114, 3.17435395130982011735776171050, 3.46679016476891576318566202243, 3.80765250014432627294164915546, 4.07195637779127095495422876586, 4.11532167147719291198302511867, 4.13412141401664839718835012810, 4.22767280869453675026258740036, 4.60100312821744041496756926978, 4.60677971329333379998433838659, 4.70502370528893507951738886305, 4.87647907383505725749952030025
Plot not available for L-functions of degree greater than 10.
