| L(s) = 1 | − 4·11-s + 16-s + 16·19-s + 24·31-s + 32·59-s − 24·61-s − 16·71-s + 48·79-s + 81-s − 48·101-s − 48·109-s + 50·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1/4·16-s + 3.67·19-s + 4.31·31-s + 4.16·59-s − 3.07·61-s − 1.89·71-s + 5.40·79-s + 1/9·81-s − 4.77·101-s − 4.59·109-s + 4.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 0.301·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.119507753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.119507753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 194 T^{4} + 171 p^{2} T^{8} - 194 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 + 96 T^{2} + 4382 T^{4} + 125760 T^{6} + 2520387 T^{8} + 125760 p^{2} T^{10} + 4382 p^{4} T^{12} + 96 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 8 T + 13 T^{2} - 104 T^{3} + 1024 T^{4} - 104 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 48 T^{2} + 2009 T^{4} + 59568 T^{6} + 1666512 T^{8} + 59568 p^{2} T^{10} + 2009 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 144 T^{2} + 11089 T^{4} - 601488 T^{6} + 25035696 T^{8} - 601488 p^{2} T^{10} + 11089 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 126 T^{2} + 7139 T^{4} - 126 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 3548 T^{4} + 6433446 T^{8} - 3548 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 144 T^{2} + 10457 T^{4} + 510480 T^{6} + 22955952 T^{8} + 510480 p^{2} T^{10} + 10457 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 5407 T^{4} + 21345168 T^{8} - 5407 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 16 T + 86 T^{2} - 832 T^{3} + 10315 T^{4} - 832 p T^{5} + 86 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 12 T + 118 T^{2} + 840 T^{3} + 4107 T^{4} + 840 p T^{5} + 118 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 7922 T^{4} + 42606963 T^{8} - 7922 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 6242 T^{4} + 10564323 T^{8} - 6242 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 24 T + 298 T^{2} - 2544 T^{3} + 20163 T^{4} - 2544 p T^{5} + 298 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 10414 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 2308 T^{4} - 29276922 T^{8} + 2308 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.32277490152149904467670307791, −4.24812951471260870745557838280, −3.87614442903920584499771060866, −3.85041360758353400398175635062, −3.83257202672997005964897031956, −3.36932458917577782962607224021, −3.32553859947262284354403362370, −3.29129735489594627087958617364, −3.25055019405868756347445494129, −3.07498675517438374035368571166, −2.88805851474339594655982260571, −2.83870651920713989607696851904, −2.51989028844556403997757285253, −2.51637017398638379648886334205, −2.40872892476992567989264722209, −2.04853582431114159954419276417, −2.01580043639252985362491514988, −2.00908150847754060703881411237, −1.37917900604480895680059677254, −1.35471732561214671786286055638, −1.09847944289844968194855711037, −1.02230801969975112304564942365, −0.962550171260421023745527618192, −0.58562055757380370034116585199, −0.25378414000441899411867544876,
0.25378414000441899411867544876, 0.58562055757380370034116585199, 0.962550171260421023745527618192, 1.02230801969975112304564942365, 1.09847944289844968194855711037, 1.35471732561214671786286055638, 1.37917900604480895680059677254, 2.00908150847754060703881411237, 2.01580043639252985362491514988, 2.04853582431114159954419276417, 2.40872892476992567989264722209, 2.51637017398638379648886334205, 2.51989028844556403997757285253, 2.83870651920713989607696851904, 2.88805851474339594655982260571, 3.07498675517438374035368571166, 3.25055019405868756347445494129, 3.29129735489594627087958617364, 3.32553859947262284354403362370, 3.36932458917577782962607224021, 3.83257202672997005964897031956, 3.85041360758353400398175635062, 3.87614442903920584499771060866, 4.24812951471260870745557838280, 4.32277490152149904467670307791
Plot not available for L-functions of degree greater than 10.
