| L(s) = 1 | − 8·43-s + 8·83-s − 8·107-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 8·43-s + 8·83-s − 8·107-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2154853339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2154853339\) |
| \(L(1)\) |
|
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 | \( 1 + T^{8} \) |
| 17 | \( 1 + T^{8} \) |
| good | 5 | \( 1 + T^{16} \) |
| 7 | \( 1 + T^{16} \) |
| 11 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( 1 + T^{16} \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( 1 + T^{16} \) |
| 41 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{8} )^{2} \) |
| 59 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 61 | \( 1 + T^{16} \) |
| 67 | \( ( 1 + T^{8} )^{2} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.21623021130690987079458257192, −5.06198050948968082724024420023, −5.03369671349619399853037110187, −4.96414542674920618453371624196, −4.70964419226433234520834084455, −4.59727183674537313206261360576, −4.57045741579681992659861905089, −4.24820270240385109453908043735, −3.86906567783537032387911895940, −3.77579477130393020563288229490, −3.76122532379670187670661762304, −3.67394816262411704870746226673, −3.65344935182624333732738392621, −3.36176885451074902688848344946, −3.07685623229264143913271079430, −2.88740142808775247994868473731, −2.87835589878174848708148280937, −2.69767315888045546575505374469, −2.38491269646857317698040132903, −2.07214209543100881357120571184, −2.05391184680140060346904513205, −1.77022214936509768958723047748, −1.56455475444913670940882748655, −1.32595840352225688512182775267, −1.11338188910639087635443663018,
1.11338188910639087635443663018, 1.32595840352225688512182775267, 1.56455475444913670940882748655, 1.77022214936509768958723047748, 2.05391184680140060346904513205, 2.07214209543100881357120571184, 2.38491269646857317698040132903, 2.69767315888045546575505374469, 2.87835589878174848708148280937, 2.88740142808775247994868473731, 3.07685623229264143913271079430, 3.36176885451074902688848344946, 3.65344935182624333732738392621, 3.67394816262411704870746226673, 3.76122532379670187670661762304, 3.77579477130393020563288229490, 3.86906567783537032387911895940, 4.24820270240385109453908043735, 4.57045741579681992659861905089, 4.59727183674537313206261360576, 4.70964419226433234520834084455, 4.96414542674920618453371624196, 5.03369671349619399853037110187, 5.06198050948968082724024420023, 5.21623021130690987079458257192
Plot not available for L-functions of degree greater than 10.
