| L(s) = 1 | − 24·11-s + 16-s + 36·29-s + 20·31-s + 12·41-s + 36·49-s − 36·59-s + 16·61-s + 12·101-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s − 24·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 7.23·11-s + 1/4·16-s + 6.68·29-s + 3.59·31-s + 1.87·41-s + 36/7·49-s − 4.68·59-s + 2.04·61-s + 1.19·101-s + 24.3·121-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 + 2.76·169-s + 0.0760·173-s − 1.80·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.912776306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.912776306\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 36 T^{2} + 634 T^{4} - 7272 T^{6} + 59571 T^{8} - 7272 p^{2} T^{10} + 634 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 36 T^{2} + 682 T^{4} - 9000 T^{6} + 106947 T^{8} - 9000 p^{2} T^{10} + 682 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + 339 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 108 T^{2} + 5818 T^{4} - 208440 T^{6} + 5501811 T^{8} - 208440 p^{2} T^{10} + 5818 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 18 T + 188 T^{2} - 1404 T^{3} + 8259 T^{4} - 1404 p T^{5} + 188 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 20 p T^{5} + 40 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 644 T^{4} - 1762266 T^{8} - 644 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 72 T^{2} + 1633 T^{4} + 6840 T^{6} - 214704 T^{8} + 6840 p^{2} T^{10} + 1633 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 508 T^{4} - 3205722 T^{8} + 508 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 18 T + 137 T^{2} + 1242 T^{3} + 12372 T^{4} + 1242 p T^{5} + 137 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 216 T^{2} + 26737 T^{4} - 2415960 T^{6} + 174766032 T^{8} - 2415960 p^{2} T^{10} + 26737 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 116 T^{2} + 12474 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 3503 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 58 T^{2} - 2877 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 432 T^{2} + 91513 T^{4} - 12659760 T^{6} + 1239875616 T^{8} - 12659760 p^{2} T^{10} + 91513 p^{4} T^{12} - 432 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 360 T^{2} + 65929 T^{4} + 8182440 T^{6} + 834546960 T^{8} + 8182440 p^{2} T^{10} + 65929 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} \) |
| 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
−4.28731146710516578712095962240, −4.26277771869616908657097817613, −3.94065396220301395671835178167, −3.41754390298525452445190795347, −3.39618747906898522597586928233, −3.32544831218366946679323630948, −3.16107108757377568312868284609, −3.15097453782845820646721446655, −2.84460850387466764855912194164, −2.80971730472912072711294065948, −2.78205479271066931751110697021, −2.64217034944461984353757988534, −2.60893717886503312234611264071, −2.35909832808079493635886552357, −2.34113740123553815283726734825, −2.29621793113475562922949223624, −1.98789598813987747062807241495, −1.80400196965886297800308886043, −1.56044899418958829376693773538, −1.16743597125724983301604010061, −0.954111221095402763950985667687, −0.77579919734473929263022979777, −0.60993680649904455083546287483, −0.56791540040831492613850238858, −0.40388732302127486146344689973,
0.40388732302127486146344689973, 0.56791540040831492613850238858, 0.60993680649904455083546287483, 0.77579919734473929263022979777, 0.954111221095402763950985667687, 1.16743597125724983301604010061, 1.56044899418958829376693773538, 1.80400196965886297800308886043, 1.98789598813987747062807241495, 2.29621793113475562922949223624, 2.34113740123553815283726734825, 2.35909832808079493635886552357, 2.60893717886503312234611264071, 2.64217034944461984353757988534, 2.78205479271066931751110697021, 2.80971730472912072711294065948, 2.84460850387466764855912194164, 3.15097453782845820646721446655, 3.16107108757377568312868284609, 3.32544831218366946679323630948, 3.39618747906898522597586928233, 3.41754390298525452445190795347, 3.94065396220301395671835178167, 4.26277771869616908657097817613, 4.28731146710516578712095962240
Plot not available for L-functions of degree greater than 10.
