L(s) = 1 | + 4·2-s + 6·4-s + 4·8-s + 3·16-s + 24·25-s + 32·29-s − 32·37-s + 96·50-s − 16·53-s + 128·58-s − 28·64-s − 128·74-s + 144·100-s − 64·106-s + 192·116-s + 48·121-s + 127-s − 72·128-s + 131-s + 137-s + 139-s − 192·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 2.82·2-s + 3·4-s + 1.41·8-s + 3/4·16-s + 24/5·25-s + 5.94·29-s − 5.26·37-s + 13.5·50-s − 2.19·53-s + 16.8·58-s − 7/2·64-s − 14.8·74-s + 72/5·100-s − 6.21·106-s + 17.8·116-s + 4.36·121-s + 0.0887·127-s − 6.36·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.7·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(33.88270090\) |
\(L(\frac12)\) |
\(\approx\) |
\(33.88270090\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 354 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 44 T^{2} + 814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 48 T^{2} + 1056 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1136 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 36 T^{2} + 870 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 28 T^{2} + 1990 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 128 T^{2} + 7296 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 56 T^{2} + 4450 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 140 T^{2} + 9286 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 32 T^{2} + 6640 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 44 T^{2} + 2926 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 156 T^{2} + 13014 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 32 T^{2} + 6496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 236 T^{2} + 25894 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 224 T^{2} + 25072 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 304 T^{2} + 38848 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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
−3.93246217242978761109293089812, −3.83627355501274882655575733580, −3.75485839554832555740473414261, −3.53646425887043738388242517555, −3.48066176705211590264681789042, −3.26748081779968369671768653610, −3.07369086066665307742814317111, −3.01788151796780653131800738279, −3.01304089291078148212495002278, −2.93381265723579301954906721011, −2.85165886021713775276410631828, −2.83113023157235481318387535669, −2.57714951350536271707406359861, −2.34055035396452087219776500703, −1.95363382364094375570057042803, −1.88255628763068660203725018144, −1.84101042599861127917761655037, −1.72041227116108906004761835962, −1.50772180543232620050629272840, −1.46536913053903763277420125619, −0.892584406168074569971257805463, −0.877742154666773418036744179240, −0.816137085359958016218680752719, −0.67469276887324311426059706139, −0.22990299786690244910795053257,
0.22990299786690244910795053257, 0.67469276887324311426059706139, 0.816137085359958016218680752719, 0.877742154666773418036744179240, 0.892584406168074569971257805463, 1.46536913053903763277420125619, 1.50772180543232620050629272840, 1.72041227116108906004761835962, 1.84101042599861127917761655037, 1.88255628763068660203725018144, 1.95363382364094375570057042803, 2.34055035396452087219776500703, 2.57714951350536271707406359861, 2.83113023157235481318387535669, 2.85165886021713775276410631828, 2.93381265723579301954906721011, 3.01304089291078148212495002278, 3.01788151796780653131800738279, 3.07369086066665307742814317111, 3.26748081779968369671768653610, 3.48066176705211590264681789042, 3.53646425887043738388242517555, 3.75485839554832555740473414261, 3.83627355501274882655575733580, 3.93246217242978761109293089812
Plot not available for L-functions of degree greater than 10.