| L(s) = 1 | + 2·3-s − 2·5-s + 5·9-s + 2·11-s − 4·13-s − 4·15-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 10·27-s + 4·29-s + 12·31-s + 4·33-s − 4·37-s − 8·39-s − 24·41-s − 24·43-s − 10·45-s + 18·47-s + 4·51-s − 16·53-s − 4·55-s + 8·57-s + 12·59-s − 8·61-s + 8·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 5/3·9-s + 0.603·11-s − 1.10·13-s − 1.03·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.92·27-s + 0.742·29-s + 2.15·31-s + 0.696·33-s − 0.657·37-s − 1.28·39-s − 3.74·41-s − 3.65·43-s − 1.49·45-s + 2.62·47-s + 0.560·51-s − 2.19·53-s − 0.539·55-s + 1.05·57-s + 1.56·59-s − 1.02·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.052238948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.052238948\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 29 T^{2} + 2 T^{3} + 732 T^{4} + 2 p T^{5} - 29 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 56 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 64 T^{2} - 216 T^{3} + 975 T^{4} - 216 p T^{5} + 64 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 36 T^{2} - 376 T^{3} - 1561 T^{4} - 376 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 151 T^{2} - 1422 T^{3} + 12492 T^{4} - 1422 p T^{5} + 151 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 88 T^{2} + 992 T^{3} + 11847 T^{4} + 992 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 216 p T^{5} + 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T - 66 T^{2} + 64 T^{3} + 9275 T^{4} + 64 p T^{5} - 66 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 132 T^{2} + 12935 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 54 T^{2} + 896 T^{3} + 16787 T^{4} + 896 p T^{5} + 54 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 98 T^{2} + 128 T^{3} + 12627 T^{4} + 128 p T^{5} - 98 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 14 T + 193 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.61911396416858915494472260593, −6.40916477521136860935782732972, −6.12301376777661502796714586192, −5.91762640976662757672093477668, −5.87687592712878200582556584380, −5.19546786715997180017672272889, −5.15807653924357763755017287994, −4.91255397785963325440714189745, −4.75909047303676119279642096932, −4.59005967740016512711309441741, −4.55937521337716759673232374726, −4.03044102888992278673656498503, −3.72241990104102566286019176269, −3.62009986146080255399130295807, −3.46664150563461534066233679401, −3.17169697004422722589404068773, −3.10240252970006382363844646452, −2.69322064924107895466724565350, −2.45120666802325773622392112246, −1.98441870594780553989992625647, −1.77289942591159686217517774889, −1.68406612118923734709058021469, −1.22936259812997770880375968335, −0.60779134447075641132921764215, −0.53218892069617798438680302840,
0.53218892069617798438680302840, 0.60779134447075641132921764215, 1.22936259812997770880375968335, 1.68406612118923734709058021469, 1.77289942591159686217517774889, 1.98441870594780553989992625647, 2.45120666802325773622392112246, 2.69322064924107895466724565350, 3.10240252970006382363844646452, 3.17169697004422722589404068773, 3.46664150563461534066233679401, 3.62009986146080255399130295807, 3.72241990104102566286019176269, 4.03044102888992278673656498503, 4.55937521337716759673232374726, 4.59005967740016512711309441741, 4.75909047303676119279642096932, 4.91255397785963325440714189745, 5.15807653924357763755017287994, 5.19546786715997180017672272889, 5.87687592712878200582556584380, 5.91762640976662757672093477668, 6.12301376777661502796714586192, 6.40916477521136860935782732972, 6.61911396416858915494472260593
