| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s + 330·16-s − 8·23-s − 792·32-s + 64·46-s + 12·49-s − 8·53-s + 1.71e3·64-s + 48·79-s − 288·92-s − 96·98-s + 64·106-s + 16·107-s + 48·113-s + 80·121-s + 127-s − 3.43e3·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 384·158-s + 163-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s + 82.5·16-s − 1.66·23-s − 140.·32-s + 9.43·46-s + 12/7·49-s − 1.09·53-s + 214.5·64-s + 5.40·79-s − 30.0·92-s − 9.69·98-s + 6.21·106-s + 1.54·107-s + 4.51·113-s + 7.27·121-s + 0.0887·127-s − 303.·128-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 − 30.5·158-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9381679553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9381679553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 11 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 22 T^{2} + 259 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 16 T^{2} - 14 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + T + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 30 T^{2} + 107 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 34 T^{2} + 1411 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 74 T^{2} + 3931 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 128 T^{2} + 7714 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 46 T^{2} + 5691 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 214 T^{2} + 18691 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 - 20 T^{2} - 2618 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 72 T^{2} + 4754 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12 T + 144 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 62 T^{2} + 9739 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 116 T^{2} + 6406 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 48 T^{2} - 9406 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 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
−3.61261201988250633661110299621, −3.19743576660060655518440465008, −3.13311808277714411674218311228, −3.12440634316891798124181334725, −3.03925044760253783128288857005, −2.83216607829951677981930043383, −2.72796970284675827340773650657, −2.69294376489547866044025259639, −2.53881126697437707771702740197, −2.18018595585389559316627035009, −2.12739128706677589010360492127, −2.10663081065998814485596459061, −1.99097960309358159841943461110, −1.93633265305773165420542068486, −1.84485130205414414187123624013, −1.59567589643450197558645628472, −1.53998293753685243949603595295, −1.37111067517537875049375871336, −1.21819689321840552581469706288, −0.76125672152594412186332497541, −0.68926245233958698564802213843, −0.66876321119493066531613599678, −0.64857798360169942970717899194, −0.37439406263420404213465104189, −0.35889591257301730401713236085,
0.35889591257301730401713236085, 0.37439406263420404213465104189, 0.64857798360169942970717899194, 0.66876321119493066531613599678, 0.68926245233958698564802213843, 0.76125672152594412186332497541, 1.21819689321840552581469706288, 1.37111067517537875049375871336, 1.53998293753685243949603595295, 1.59567589643450197558645628472, 1.84485130205414414187123624013, 1.93633265305773165420542068486, 1.99097960309358159841943461110, 2.10663081065998814485596459061, 2.12739128706677589010360492127, 2.18018595585389559316627035009, 2.53881126697437707771702740197, 2.69294376489547866044025259639, 2.72796970284675827340773650657, 2.83216607829951677981930043383, 3.03925044760253783128288857005, 3.12440634316891798124181334725, 3.13311808277714411674218311228, 3.19743576660060655518440465008, 3.61261201988250633661110299621
Plot not available for L-functions of degree greater than 10.
