| L(s) = 1 | − 8·9-s − 16·23-s + 16·31-s − 16·41-s − 24·47-s − 24·49-s − 16·71-s − 24·79-s + 32·81-s − 32·89-s + 40·97-s − 48·103-s + 48·113-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 3.33·23-s + 2.87·31-s − 2.49·41-s − 3.50·47-s − 3.42·49-s − 1.89·71-s − 2.70·79-s + 32/9·81-s − 3.39·89-s + 4.06·97-s − 4.72·103-s + 4.51·113-s − 2.18·121-s + 0.0887·127-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 + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + 48 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4\times C_2$ | \( 1 + 24 T^{2} + 288 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 48 T^{2} + 912 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 24 T^{2} + 768 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 96 T^{2} + 3984 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 144 T^{2} + 7920 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 + 72 T^{2} + 4416 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 144 T^{2} + 9744 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 168 T^{2} + 13920 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 240 T^{2} + 21840 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 216 T^{2} + 20544 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 140 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 264 T^{2} + 31104 T^{4} + 264 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.34319939907105477512475167135, −5.93597678658563116646136720519, −5.91903773291309565009646741703, −5.89434665074686214914155942912, −5.86286955903026277753706581164, −5.20933732509306442561915567242, −5.17911307927478651227543517645, −4.93443793719623691205097567814, −4.89980640084712586872231038132, −4.61638925114882091987046423988, −4.35445396800044677249250071371, −4.22824301365795004330795202641, −3.92985835947114477429553093006, −3.61832707663481362353358112719, −3.28803327478867586933481297161, −3.26739523176804123428988061959, −3.17176982869938136850114005872, −2.94277213532426302805159272212, −2.52876896198213592239488826707, −2.36765915439789321270198716906, −2.24972520209923187294074008579, −1.88099880511095931526334869756, −1.41548776858370716236933326977, −1.36748528969237929528614510727, −1.18687434518358345282297487856, 0, 0, 0, 0,
1.18687434518358345282297487856, 1.36748528969237929528614510727, 1.41548776858370716236933326977, 1.88099880511095931526334869756, 2.24972520209923187294074008579, 2.36765915439789321270198716906, 2.52876896198213592239488826707, 2.94277213532426302805159272212, 3.17176982869938136850114005872, 3.26739523176804123428988061959, 3.28803327478867586933481297161, 3.61832707663481362353358112719, 3.92985835947114477429553093006, 4.22824301365795004330795202641, 4.35445396800044677249250071371, 4.61638925114882091987046423988, 4.89980640084712586872231038132, 4.93443793719623691205097567814, 5.17911307927478651227543517645, 5.20933732509306442561915567242, 5.86286955903026277753706581164, 5.89434665074686214914155942912, 5.91903773291309565009646741703, 5.93597678658563116646136720519, 6.34319939907105477512475167135
