| L(s) = 1 | − 2·3-s + 5·9-s − 20·17-s + 10·23-s + 14·25-s − 2·27-s + 20·29-s − 18·43-s − 19·49-s + 40·51-s + 52·53-s − 4·61-s − 20·69-s − 28·75-s − 32·79-s + 12·81-s − 40·87-s − 12·101-s − 48·103-s − 10·107-s − 4·113-s − 35·121-s + 127-s + 36·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 5/3·9-s − 4.85·17-s + 2.08·23-s + 14/5·25-s − 0.384·27-s + 3.71·29-s − 2.74·43-s − 2.71·49-s + 5.60·51-s + 7.14·53-s − 0.512·61-s − 2.40·69-s − 3.23·75-s − 3.60·79-s + 4/3·81-s − 4.28·87-s − 1.19·101-s − 4.72·103-s − 0.966·107-s − 0.376·113-s − 3.18·121-s + 0.0887·127-s + 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7500084390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7500084390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 + T - T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 19 T^{2} + 177 T^{4} + 1634 T^{6} + 13766 T^{8} + 1634 p^{2} T^{10} + 177 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 35 T^{2} + 681 T^{4} + 10570 T^{6} + 133070 T^{8} + 10570 p^{2} T^{10} + 681 p^{4} T^{12} + 35 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 + 67 T^{2} + 2649 T^{4} + 74906 T^{6} + 1634750 T^{8} + 74906 p^{2} T^{10} + 2649 p^{4} T^{12} + 67 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 5 T + 11 T^{2} + 160 T^{3} - 908 T^{4} + 160 p T^{5} + 11 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | \( 1 + 26 T^{2} - 63 p T^{4} - 2678 T^{6} + 6658964 T^{8} - 2678 p^{2} T^{10} - 63 p^{5} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 9 T - 21 T^{2} + 144 T^{3} + 5072 T^{4} + 144 p T^{5} - 21 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 + 227 T^{2} + 31689 T^{4} + 2923306 T^{6} + 202135790 T^{8} + 2923306 p^{2} T^{10} + 31689 p^{4} T^{12} + 227 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 2 T - 51 T^{2} - 134 T^{3} - 940 T^{4} - 134 p T^{5} - 51 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 5 T^{2} - 6711 T^{4} + 11210 T^{6} + 25110350 T^{8} + 11210 p^{2} T^{10} - 6711 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 + 203 T^{2} + 21169 T^{4} + 2021474 T^{6} + 168455350 T^{8} + 2021474 p^{2} T^{10} + 21169 p^{4} T^{12} + 203 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 103 T^{2} + 4704 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 124 T^{2} + 7830 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 255 T^{2} + 34801 T^{4} + 3667410 T^{6} + 335684910 T^{8} + 3667410 p^{2} T^{10} + 34801 p^{4} T^{12} + 255 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 + 63 T^{2} + 8065 T^{4} - 1443582 T^{6} - 109096386 T^{8} - 1443582 p^{2} T^{10} + 8065 p^{4} T^{12} + 63 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.28127947334557937436318880837, −4.00714021138735409489931529560, −3.91386117800414127998395683228, −3.74582452348395738655745793709, −3.65257749889132477004877918965, −3.52996585734844966889662169558, −3.16747372757871725369622688526, −3.04084058216161408047456724508, −2.96931927315382496678766940386, −2.84008880530730141856658389543, −2.76154127550521019090614588291, −2.70679949368567254106977845223, −2.49786221374209368126520620322, −2.35923642400027690471684041993, −2.21410168098523973987471451964, −2.06958132103718470803977403720, −1.87413311894180200697507495936, −1.48026807406248715713542260710, −1.45308493977658499567647898229, −1.33512719820871583694866093276, −1.12175117746517501257456016481, −0.934972108495174345304872610848, −0.73801506733963459432371914337, −0.48036939459487722997377192541, −0.11311401418781187692683725998,
0.11311401418781187692683725998, 0.48036939459487722997377192541, 0.73801506733963459432371914337, 0.934972108495174345304872610848, 1.12175117746517501257456016481, 1.33512719820871583694866093276, 1.45308493977658499567647898229, 1.48026807406248715713542260710, 1.87413311894180200697507495936, 2.06958132103718470803977403720, 2.21410168098523973987471451964, 2.35923642400027690471684041993, 2.49786221374209368126520620322, 2.70679949368567254106977845223, 2.76154127550521019090614588291, 2.84008880530730141856658389543, 2.96931927315382496678766940386, 3.04084058216161408047456724508, 3.16747372757871725369622688526, 3.52996585734844966889662169558, 3.65257749889132477004877918965, 3.74582452348395738655745793709, 3.91386117800414127998395683228, 4.00714021138735409489931529560, 4.28127947334557937436318880837
Plot not available for L-functions of degree greater than 10.
