| L(s) = 1 | − 4·7-s + 24·17-s + 12·23-s − 24·41-s − 12·47-s + 8·49-s − 32·73-s + 96·79-s − 8·81-s − 32·97-s + 28·103-s − 96·119-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 5.82·17-s + 2.50·23-s − 3.74·41-s − 1.75·47-s + 8/7·49-s − 3.74·73-s + 10.8·79-s − 8/9·81-s − 3.24·97-s + 2.75·103-s − 8.80·119-s − 4.36·121-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 − 3.78·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.806275293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.806275293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 3 | \( 1 + 8 T^{4} + 94 T^{8} + 8 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 82 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 302 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 - 6 T + 18 T^{2} - 102 T^{3} + 542 T^{4} - 102 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 64 T^{2} + 2190 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2338 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 2632 T^{4} + 3107358 T^{8} + 2632 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 6 T + 18 T^{2} + 246 T^{3} + 3326 T^{4} + 246 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2846 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 112 T^{2} + 9822 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 56 T^{4} - 39549474 T^{8} - 56 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 80 T^{2} + 4878 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 7838 T^{4} + 1008 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 8200 T^{4} + 98353758 T^{8} + 8200 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 116 T^{2} + 7110 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 15038 T^{4} + 1392 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.13438410011949035827611955586, −5.00300328504119628250658705528, −4.99797413585837926245795086769, −4.97633324809786849724108151787, −4.84907233032674634405910677312, −4.61285552829091278501594235652, −4.23458053509013375024023428511, −3.89852017623922189853597685217, −3.85544204918237329699425067864, −3.75433268999048296284672652914, −3.50072910890815620855250866064, −3.43520546188331798267294970685, −3.35546001902201512383705057784, −3.27082878960383257427154112392, −3.02196512345690803106487342514, −2.87327222323351124839965407479, −2.83380114410836083869967203692, −2.42332715057219706725686221249, −2.20688305333237280610987585714, −1.92554315144311601114371901541, −1.56856732259898553235872835770, −1.36439592433597741770647535212, −1.07028622373396869751093103667, −1.04673497159672659432434930475, −0.45278802364928996872331092140,
0.45278802364928996872331092140, 1.04673497159672659432434930475, 1.07028622373396869751093103667, 1.36439592433597741770647535212, 1.56856732259898553235872835770, 1.92554315144311601114371901541, 2.20688305333237280610987585714, 2.42332715057219706725686221249, 2.83380114410836083869967203692, 2.87327222323351124839965407479, 3.02196512345690803106487342514, 3.27082878960383257427154112392, 3.35546001902201512383705057784, 3.43520546188331798267294970685, 3.50072910890815620855250866064, 3.75433268999048296284672652914, 3.85544204918237329699425067864, 3.89852017623922189853597685217, 4.23458053509013375024023428511, 4.61285552829091278501594235652, 4.84907233032674634405910677312, 4.97633324809786849724108151787, 4.99797413585837926245795086769, 5.00300328504119628250658705528, 5.13438410011949035827611955586
Plot not available for L-functions of degree greater than 10.
