| L(s) = 1 | − 2·2-s + 2·4-s − 4·8-s + 8·16-s − 8·31-s − 8·32-s − 8·37-s + 8·41-s + 28·43-s + 20·49-s + 32·53-s + 16·62-s + 8·64-s + 36·67-s − 8·71-s + 16·74-s + 32·79-s − 16·82-s + 12·83-s − 56·86-s + 8·89-s − 40·98-s − 64·106-s − 4·107-s + 36·121-s − 16·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.41·8-s + 2·16-s − 1.43·31-s − 1.41·32-s − 1.31·37-s + 1.24·41-s + 4.26·43-s + 20/7·49-s + 4.39·53-s + 2.03·62-s + 64-s + 4.39·67-s − 0.949·71-s + 1.85·74-s + 3.60·79-s − 1.76·82-s + 1.31·83-s − 6.03·86-s + 0.847·89-s − 4.04·98-s − 6.21·106-s − 0.386·107-s + 3.27·121-s − 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.167048193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167048193\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1274 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.80132734858067526401386003109, −6.32290512061017416769039411460, −6.21577791582568621600377915153, −6.01679419007930889797554109180, −5.77019656136358893423851221171, −5.52252923029711317383415680308, −5.42194278750392307123297804497, −5.28614684557470177210932685509, −5.07421026850103707576951762632, −4.52498286377547978460397197121, −4.43065472863343550565325607578, −3.98609820394365710643385744079, −3.93661535712193837689058296910, −3.65479901934812892995093085235, −3.47856842984895355613319264577, −3.39152540294231288304865650048, −2.55100191328753643492725538487, −2.54325279081325834277249328822, −2.44817841408723232151916021635, −2.11488947378262746748655621640, −2.04677501806260504717959086173, −1.18688230191647644284742036211, −0.957815159753861520918419868526, −0.64054266283646161527207796357, −0.58206049848780155598991877222,
0.58206049848780155598991877222, 0.64054266283646161527207796357, 0.957815159753861520918419868526, 1.18688230191647644284742036211, 2.04677501806260504717959086173, 2.11488947378262746748655621640, 2.44817841408723232151916021635, 2.54325279081325834277249328822, 2.55100191328753643492725538487, 3.39152540294231288304865650048, 3.47856842984895355613319264577, 3.65479901934812892995093085235, 3.93661535712193837689058296910, 3.98609820394365710643385744079, 4.43065472863343550565325607578, 4.52498286377547978460397197121, 5.07421026850103707576951762632, 5.28614684557470177210932685509, 5.42194278750392307123297804497, 5.52252923029711317383415680308, 5.77019656136358893423851221171, 6.01679419007930889797554109180, 6.21577791582568621600377915153, 6.32290512061017416769039411460, 6.80132734858067526401386003109
