| L(s) = 1 | − 2·3-s − 4·4-s + 7·9-s + 8·12-s + 32·13-s + 12·16-s + 4·19-s + 62·25-s − 38·27-s + 20·31-s − 28·36-s − 4·37-s − 64·39-s + 104·43-s − 24·48-s − 128·52-s − 8·57-s + 212·61-s − 32·64-s − 156·67-s + 132·73-s − 124·75-s − 16·76-s + 52·79-s + 52·81-s − 40·93-s − 288·97-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 4-s + 7/9·9-s + 2/3·12-s + 2.46·13-s + 3/4·16-s + 4/19·19-s + 2.47·25-s − 1.40·27-s + 0.645·31-s − 7/9·36-s − 0.108·37-s − 1.64·39-s + 2.41·43-s − 1/2·48-s − 2.46·52-s − 0.140·57-s + 3.47·61-s − 1/2·64-s − 2.32·67-s + 1.80·73-s − 1.65·75-s − 0.210·76-s + 0.658·79-s + 0.641·81-s − 0.430·93-s − 2.96·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.877555272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.877555272\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T - p T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1859 T^{4} - 62 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 70 T^{2} + 28307 T^{4} + 70 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 314 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1118 T^{2} + 479171 T^{4} - 1118 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 525 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1370 T^{2} + 921107 T^{4} - 1370 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 764 T^{2} + 680486 T^{4} - 764 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 1397 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5788 T^{2} + 13912710 T^{4} - 5788 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 52 T + 3582 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 538 T^{2} + 8227923 T^{4} - 538 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 5854 T^{2} + 23321859 T^{4} - 5854 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 746 T^{2} + 11330051 T^{4} - 746 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 106 T + 9899 T^{2} - 106 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 78 T + 6781 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1316 T^{2} + 44745734 T^{4} - 1316 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 66 T + 11395 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 26 T + 6293 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 18404 T^{2} + 168689894 T^{4} - 18404 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2542 T^{2} + 2901651 T^{4} - 2542 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 144 T + 21802 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.404935580268870369818579193244, −8.255235512952266081766217015138, −7.79294444064856523545935606616, −7.69306313301148348378270484487, −7.21214510340235141244244742727, −6.95441419509550204842714320182, −6.82526076093372467992223405148, −6.39909797431768254802998202256, −6.25319823952313739396282555149, −5.96243519900633720155562442775, −5.63138937584765816629117149087, −5.29618952767349512182379637897, −5.18314289137193213584127821693, −4.94812472664012014251989030849, −4.26089074644920909922743265941, −4.11706371382500959837952338875, −4.07976530946999687676172405135, −3.72320486076070404586938985314, −3.19178264291594637127269547048, −2.88705829099593044375638349506, −2.51611386598418328229211040475, −1.76622634024379324917685443748, −1.29680317033067065065400266907, −0.921115683599993319628590636427, −0.58954289468100479647299394962,
0.58954289468100479647299394962, 0.921115683599993319628590636427, 1.29680317033067065065400266907, 1.76622634024379324917685443748, 2.51611386598418328229211040475, 2.88705829099593044375638349506, 3.19178264291594637127269547048, 3.72320486076070404586938985314, 4.07976530946999687676172405135, 4.11706371382500959837952338875, 4.26089074644920909922743265941, 4.94812472664012014251989030849, 5.18314289137193213584127821693, 5.29618952767349512182379637897, 5.63138937584765816629117149087, 5.96243519900633720155562442775, 6.25319823952313739396282555149, 6.39909797431768254802998202256, 6.82526076093372467992223405148, 6.95441419509550204842714320182, 7.21214510340235141244244742727, 7.69306313301148348378270484487, 7.79294444064856523545935606616, 8.255235512952266081766217015138, 8.404935580268870369818579193244
