| L(s) = 1 | − 2·4-s + 8·9-s + 8·11-s + 3·16-s + 4·19-s + 12·29-s + 4·31-s − 16·36-s + 4·41-s − 16·44-s + 26·49-s + 16·59-s − 28·61-s − 4·64-s + 8·71-s − 8·76-s + 28·79-s + 30·81-s + 8·89-s + 64·99-s + 32·109-s − 24·116-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 8/3·9-s + 2.41·11-s + 3/4·16-s + 0.917·19-s + 2.22·29-s + 0.718·31-s − 8/3·36-s + 0.624·41-s − 2.41·44-s + 26/7·49-s + 2.08·59-s − 3.58·61-s − 1/2·64-s + 0.949·71-s − 0.917·76-s + 3.15·79-s + 10/3·81-s + 0.847·89-s + 6.43·99-s + 3.06·109-s − 2.22·116-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.892282593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.892282593\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 595 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 14 p T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 214 T^{2} + 19779 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 206 T^{2} + 19467 T^{4} - 206 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 205 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 30166 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 132 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 17650 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.62039233305205583814758845546, −6.22978856789006981353203572928, −6.09435630319836451380977712301, −5.95104896992916492072495662003, −5.46550311961132956460317594584, −5.44695590502583191186171774230, −5.17202779541679612564311019916, −4.73983699755431962981262033099, −4.60433949139618820413667525746, −4.47962465994249505892367102207, −4.47713048544575798849074410149, −4.03774999840706421564401706946, −3.95240015049868268979877818359, −3.70883091445091825108715958167, −3.54547702680912995508425612518, −3.24844350354251007022995944677, −2.97111457244028617562885861556, −2.59514604825645346988188738873, −2.18378170154514213989980798405, −2.06806237120489962487702326854, −1.66696069816880308116917563446, −1.16458072465736565648309825773, −1.13736093972284511768033167051, −0.848641541044952427111408936370, −0.68185035125219129401295522645,
0.68185035125219129401295522645, 0.848641541044952427111408936370, 1.13736093972284511768033167051, 1.16458072465736565648309825773, 1.66696069816880308116917563446, 2.06806237120489962487702326854, 2.18378170154514213989980798405, 2.59514604825645346988188738873, 2.97111457244028617562885861556, 3.24844350354251007022995944677, 3.54547702680912995508425612518, 3.70883091445091825108715958167, 3.95240015049868268979877818359, 4.03774999840706421564401706946, 4.47713048544575798849074410149, 4.47962465994249505892367102207, 4.60433949139618820413667525746, 4.73983699755431962981262033099, 5.17202779541679612564311019916, 5.44695590502583191186171774230, 5.46550311961132956460317594584, 5.95104896992916492072495662003, 6.09435630319836451380977712301, 6.22978856789006981353203572928, 6.62039233305205583814758845546
