| L(s) = 1 | − 4·3-s + 6·4-s − 24·12-s + 17·16-s + 4·17-s + 20·23-s − 4·25-s + 24·27-s + 16·29-s + 4·43-s − 68·48-s + 16·49-s − 16·51-s − 24·53-s + 56·61-s + 30·64-s + 24·68-s − 80·69-s + 16·75-s − 16·79-s − 42·81-s − 64·87-s + 120·92-s − 24·100-s + 8·101-s + 104·103-s − 68·107-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3·4-s − 6.92·12-s + 17/4·16-s + 0.970·17-s + 4.17·23-s − 4/5·25-s + 4.61·27-s + 2.97·29-s + 0.609·43-s − 9.81·48-s + 16/7·49-s − 2.24·51-s − 3.29·53-s + 7.17·61-s + 15/4·64-s + 2.91·68-s − 9.63·69-s + 1.84·75-s − 1.80·79-s − 4.66·81-s − 6.86·87-s + 12.5·92-s − 2.39·100-s + 0.796·101-s + 10.2·103-s − 6.57·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.434095675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.434095675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T + 5 T^{2} + p T^{3} - 11 T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{5} T^{7} + p^{4} T^{8} )( 1 + p^{2} T + 5 T^{2} - p T^{3} - 11 T^{4} - p^{2} T^{5} + 5 p^{2} T^{6} + p^{5} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( ( 1 + 2 T + 2 p T^{2} + 8 T^{3} + 19 T^{4} + 8 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 16 T^{2} + 130 T^{4} - 928 T^{6} + 6547 T^{8} - 928 p^{2} T^{10} + 130 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 14 T^{2} + 9 p T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 2 T + 50 T^{2} - 92 T^{3} + 1135 T^{4} - 92 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 p T^{2} + 891 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 10 T + 98 T^{2} - 544 T^{3} + 137 p T^{4} - 544 p T^{5} + 98 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 8 T + 98 T^{2} - 656 T^{3} + 4003 T^{4} - 656 p T^{5} + 98 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 184 T^{2} + 16234 T^{4} - 944464 T^{6} + 40333627 T^{8} - 944464 p^{2} T^{10} + 16234 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 150 T^{2} + 8939 T^{4} - 150 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 2 T + 154 T^{2} - 248 T^{3} + 9559 T^{4} - 248 p T^{5} + 154 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 168 T^{2} + 17500 T^{4} - 1260696 T^{6} + 68079174 T^{8} - 1260696 p^{2} T^{10} + 17500 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 12 T + 248 T^{2} + 36 p T^{3} + 20622 T^{4} + 36 p^{2} T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 388 T^{2} + 68338 T^{4} - 7257856 T^{6} + 515797243 T^{8} - 7257856 p^{2} T^{10} + 68338 p^{4} T^{12} - 388 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 28 T + 502 T^{2} - 6088 T^{3} + 55063 T^{4} - 6088 p T^{5} + 502 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 296 T^{2} + 48970 T^{4} - 5314928 T^{6} + 417964507 T^{8} - 5314928 p^{2} T^{10} + 48970 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 132 T^{2} + 23890 T^{4} - 1939104 T^{6} + 187926459 T^{8} - 1939104 p^{2} T^{10} + 23890 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 352 T^{2} + 64540 T^{4} - 7783072 T^{6} + 668463622 T^{8} - 7783072 p^{2} T^{10} + 64540 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 472 T^{2} + 104380 T^{4} - 14464552 T^{6} + 1407855142 T^{8} - 14464552 p^{2} T^{10} + 104380 p^{4} T^{12} - 472 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 100 T^{2} + 18994 T^{4} - 2155360 T^{6} + 176636059 T^{8} - 2155360 p^{2} T^{10} + 18994 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 592 T^{2} + 162082 T^{4} - 27299296 T^{6} + 3149746867 T^{8} - 27299296 p^{2} T^{10} + 162082 p^{4} T^{12} - 592 p^{6} T^{14} + p^{8} T^{16} \) |
| 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
−4.43891496884894719143400793679, −4.38185973073480649636914480301, −4.05487596131246268194991160110, −3.98444516613066405255947656654, −3.88062029633142284794177026169, −3.66288863821351000791840156308, −3.53511168831663447872343064537, −3.25563553938612333802451898929, −3.21533813449926972321094078014, −3.05854680870113004097050015692, −2.92739637708683345206152690349, −2.90692093427453831961686543445, −2.68411476358167449909968385957, −2.49071663220357356569433124822, −2.44188132477319053669433571382, −2.42885529897428974126986166635, −1.96303641288524439592703328617, −1.94379476364255415255890412747, −1.81961246610896341326172451469, −1.29782475790917684072192055650, −1.25239945911159375930312235016, −0.934345043060737349453222619571, −0.885858634118713048739602333207, −0.59575703449573000515754443669, −0.45988184234573383236156157737,
0.45988184234573383236156157737, 0.59575703449573000515754443669, 0.885858634118713048739602333207, 0.934345043060737349453222619571, 1.25239945911159375930312235016, 1.29782475790917684072192055650, 1.81961246610896341326172451469, 1.94379476364255415255890412747, 1.96303641288524439592703328617, 2.42885529897428974126986166635, 2.44188132477319053669433571382, 2.49071663220357356569433124822, 2.68411476358167449909968385957, 2.90692093427453831961686543445, 2.92739637708683345206152690349, 3.05854680870113004097050015692, 3.21533813449926972321094078014, 3.25563553938612333802451898929, 3.53511168831663447872343064537, 3.66288863821351000791840156308, 3.88062029633142284794177026169, 3.98444516613066405255947656654, 4.05487596131246268194991160110, 4.38185973073480649636914480301, 4.43891496884894719143400793679
Plot not available for L-functions of degree greater than 10.
