| L(s) = 1 | − 2·3-s + 5·9-s − 2·17-s − 32·23-s + 14·25-s − 2·27-s + 8·29-s + 30·43-s − 19·49-s + 4·51-s − 8·53-s − 28·61-s + 64·69-s − 28·75-s + 64·79-s + 12·81-s − 16·87-s − 12·101-s + 48·103-s − 16·107-s + 8·113-s − 8·121-s + 127-s − 60·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 5/3·9-s − 0.485·17-s − 6.67·23-s + 14/5·25-s − 0.384·27-s + 1.48·29-s + 4.57·43-s − 2.71·49-s + 0.560·51-s − 1.09·53-s − 3.58·61-s + 7.70·69-s − 3.23·75-s + 7.20·79-s + 4/3·81-s − 1.71·87-s − 1.19·101-s + 4.72·103-s − 1.54·107-s + 0.752·113-s − 0.727·121-s + 0.0887·127-s − 5.28·129-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
| \(L(1)\) |
\(\approx\) |
\(5.924143940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.924143940\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 8 T^{2} - 126 T^{4} - 416 T^{6} + 13715 T^{8} - 416 p^{2} T^{10} - 126 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + T + 5 T^{2} - 38 T^{3} - 290 T^{4} - 38 p T^{5} + 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 40 T^{2} + 546 T^{4} + 13280 T^{6} + 402995 T^{8} + 13280 p^{2} T^{10} + 546 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 + 47 T^{2} + 793 T^{4} - 62134 T^{6} - 2893226 T^{8} - 62134 p^{2} T^{10} + 793 p^{4} T^{12} + 47 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 + 128 T^{2} + 8994 T^{4} + 515584 T^{6} + 24244835 T^{8} + 515584 p^{2} T^{10} + 8994 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 15 T + 87 T^{2} - 780 T^{3} + 7520 T^{4} - 780 p T^{5} + 87 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 27 T^{2} - 1864 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 + 200 T^{2} + 23106 T^{4} + 1986400 T^{6} + 134049875 T^{8} + 1986400 p^{2} T^{10} + 23106 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 14 T + 42 T^{2} + 448 T^{3} + 8039 T^{4} + 448 p T^{5} + 42 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 232 T^{2} + 31458 T^{4} + 3106016 T^{6} + 238470899 T^{8} + 3106016 p^{2} T^{10} + 31458 p^{4} T^{12} + 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 124 T^{2} + 7830 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | \( ( 1 + 126 T^{2} + 6467 T^{4} + 126 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 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
−3.97680483379128826602630211616, −3.92289021672047306418697473994, −3.92165128329256674956805734329, −3.86665859111340152938332840661, −3.67971844730658525302605052109, −3.57252766152847904721433033291, −3.43719726170426927696215334489, −2.99252871270861561782677401895, −2.91589934123392649958531358213, −2.80571654740627372587589775938, −2.78542316297166651237670568866, −2.75629882168115047063171897738, −2.64382028367707775391884258719, −2.08083263803870191066603440386, −2.01019100074603563484998375415, −1.99355157411813289667424416133, −1.84767745486982007947944245636, −1.77251003655047548524769878576, −1.70764084499800659365037450905, −1.27560761691022696021157421828, −1.13739928461083132435550556852, −0.66272236145869539760168595234, −0.64541948068779376037805825399, −0.56510879187932702732931013774, −0.38462669819760237456237168239,
0.38462669819760237456237168239, 0.56510879187932702732931013774, 0.64541948068779376037805825399, 0.66272236145869539760168595234, 1.13739928461083132435550556852, 1.27560761691022696021157421828, 1.70764084499800659365037450905, 1.77251003655047548524769878576, 1.84767745486982007947944245636, 1.99355157411813289667424416133, 2.01019100074603563484998375415, 2.08083263803870191066603440386, 2.64382028367707775391884258719, 2.75629882168115047063171897738, 2.78542316297166651237670568866, 2.80571654740627372587589775938, 2.91589934123392649958531358213, 2.99252871270861561782677401895, 3.43719726170426927696215334489, 3.57252766152847904721433033291, 3.67971844730658525302605052109, 3.86665859111340152938332840661, 3.92165128329256674956805734329, 3.92289021672047306418697473994, 3.97680483379128826602630211616
Plot not available for L-functions of degree greater than 10.
