| L(s) = 1 | + 2·9-s − 4·16-s + 2·25-s + 4·29-s + 2·49-s + 2·81-s + 4·107-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2·9-s − 4·16-s + 2·25-s + 4·29-s + 2·49-s + 2·81-s + 4·107-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9482674629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9482674629\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( ( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T^{4} )^{4} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 5 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 7 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 11 | \( ( 1 + T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{2} )^{8} \) |
| 19 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 23 | \( ( 1 + T^{8} )^{2} \) |
| 29 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 31 | \( ( 1 + T^{8} )^{2} \) |
| 37 | \( ( 1 + T^{8} )^{2} \) |
| 41 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{2} )^{8} \) |
| 53 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{8} )^{2} \) |
| 67 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 71 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 73 | \( ( 1 + T^{8} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{2} )^{8} \) |
| 97 | \( ( 1 + T^{8} )^{2} \) |
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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
−4.47626854888907079224476195224, −4.32630776153482582256651059005, −4.30081262195819464610008005932, −4.27239096081696410044874685171, −4.22662245441463389010899896073, −4.07944198333210381489447704871, −3.68395362707258442720718035022, −3.60384124086231015910307526542, −3.49898946140986156130608924306, −3.23826115360560689386795359322, −3.13620918918645631758983792906, −2.98667162767181594121506809039, −2.92714621327352683400653285945, −2.91875205153366249179164225574, −2.35958669733699354717631568660, −2.33880002591488946265629180735, −2.28833826810435930840556288376, −2.14175756377514327769747975781, −2.10607686371703698311108035792, −2.00852412188267381094325209538, −1.49948125710514545197710613189, −1.27293537117150382696958656217, −1.11577467580518840390409549173, −0.969756421689913246806934083400, −0.853724805332220323221377352745,
0.853724805332220323221377352745, 0.969756421689913246806934083400, 1.11577467580518840390409549173, 1.27293537117150382696958656217, 1.49948125710514545197710613189, 2.00852412188267381094325209538, 2.10607686371703698311108035792, 2.14175756377514327769747975781, 2.28833826810435930840556288376, 2.33880002591488946265629180735, 2.35958669733699354717631568660, 2.91875205153366249179164225574, 2.92714621327352683400653285945, 2.98667162767181594121506809039, 3.13620918918645631758983792906, 3.23826115360560689386795359322, 3.49898946140986156130608924306, 3.60384124086231015910307526542, 3.68395362707258442720718035022, 4.07944198333210381489447704871, 4.22662245441463389010899896073, 4.27239096081696410044874685171, 4.30081262195819464610008005932, 4.32630776153482582256651059005, 4.47626854888907079224476195224
Plot not available for L-functions of degree greater than 10.
