| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 9-s − 4·13-s − 4·16-s + 2·17-s − 2·18-s − 2·25-s − 8·26-s − 2·32-s + 4·34-s − 36-s − 4·50-s − 4·52-s + 2·53-s + 3·64-s + 2·68-s + 2·72-s + 81-s + 2·89-s − 2·100-s − 4·101-s + 8·104-s + 4·106-s + 4·117-s − 121-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 9-s − 4·13-s − 4·16-s + 2·17-s − 2·18-s − 2·25-s − 8·26-s − 2·32-s + 4·34-s − 36-s − 4·50-s − 4·52-s + 2·53-s + 3·64-s + 2·68-s + 2·72-s + 81-s + 2·89-s − 2·100-s − 4·101-s + 8·104-s + 4·106-s + 4·117-s − 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4303603343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4303603343\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.23831461657749057877131046693, −5.86073930046790263470208205370, −5.77427487029256289685927859270, −5.46422652052051979019744344082, −5.40841349069562449101373985487, −5.29216042312818692942098105083, −5.16465413115236026981141294934, −5.03876331686918251609013498655, −4.65597166750579192239009583212, −4.41065709988516126275875600245, −4.32747359206574185109780906838, −4.22647053183362401640695530634, −3.79522468698542921335865643707, −3.55395075999192802679726592975, −3.53857309532607205114071896618, −3.17695040322370612507516482751, −3.07380540530113312807222696984, −2.57648320240452858421920293901, −2.52473550059152230458130804148, −2.34984408453994711737628692473, −2.31791401184841372643827240235, −1.83085242250564752240140676837, −1.31824153352001322417227971608, −0.879482925229121803029960749735, −0.19625472632838879179832752607,
0.19625472632838879179832752607, 0.879482925229121803029960749735, 1.31824153352001322417227971608, 1.83085242250564752240140676837, 2.31791401184841372643827240235, 2.34984408453994711737628692473, 2.52473550059152230458130804148, 2.57648320240452858421920293901, 3.07380540530113312807222696984, 3.17695040322370612507516482751, 3.53857309532607205114071896618, 3.55395075999192802679726592975, 3.79522468698542921335865643707, 4.22647053183362401640695530634, 4.32747359206574185109780906838, 4.41065709988516126275875600245, 4.65597166750579192239009583212, 5.03876331686918251609013498655, 5.16465413115236026981141294934, 5.29216042312818692942098105083, 5.40841349069562449101373985487, 5.46422652052051979019744344082, 5.77427487029256289685927859270, 5.86073930046790263470208205370, 6.23831461657749057877131046693
