| L(s) = 1 | + 2·2-s + 3·4-s + 4·5-s + 2·8-s + 4·9-s + 8·10-s − 4·11-s − 14·13-s + 6·17-s + 8·18-s − 12·19-s + 12·20-s − 8·22-s + 6·25-s − 28·26-s − 14·29-s + 16·31-s − 6·32-s + 12·34-s + 12·36-s + 2·37-s − 24·38-s + 8·40-s − 6·41-s − 4·43-s − 12·44-s + 16·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.78·5-s + 0.707·8-s + 4/3·9-s + 2.52·10-s − 1.20·11-s − 3.88·13-s + 1.45·17-s + 1.88·18-s − 2.75·19-s + 2.68·20-s − 1.70·22-s + 6/5·25-s − 5.49·26-s − 2.59·29-s + 2.87·31-s − 1.06·32-s + 2.05·34-s + 2·36-s + 0.328·37-s − 3.89·38-s + 1.26·40-s − 0.937·41-s − 0.609·43-s − 1.80·44-s + 2.38·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.005188026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.005188026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 44 T^{2} + 1407 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 574 T^{3} + 3276 T^{4} + 574 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 76 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 47 T^{2} + 6 T^{3} + 3732 T^{4} + 6 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 72 T^{2} + 8 T^{3} + 5207 T^{4} + 8 p T^{5} - 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 216 p T^{5} + 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 14 T + 33 T^{2} + 574 T^{3} + 10892 T^{4} + 574 p T^{5} + 33 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 116 T^{2} + 8967 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 2 T^{2} - 48 T^{3} + 6051 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 139 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} - 896 T^{3} - 9261 T^{4} - 896 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16 T + 6 T^{2} - 896 T^{3} + 28259 T^{4} - 896 p T^{5} + 6 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.43260939826399787141804643430, −7.36451737080328065689132144795, −7.16419553915918597119040530313, −6.85319656408317174363087387904, −6.52385759139164101756742543078, −6.47814749218052134902352559733, −6.10673235977313205134280288605, −5.73087250125819981732741227793, −5.68566513896079986789412405246, −5.60639579179183138131763595348, −5.13388686821245546767550560501, −4.80169613690520254540019697438, −4.72048933824831001552265034074, −4.52509322663936736857387353690, −4.48684567522907652550270173485, −3.72873554234885787896417859087, −3.72195605877960860319738002971, −3.09871676134091068327887002135, −2.92011694089387734672592693340, −2.40791628965121693569864741095, −2.36355300339078329178547703560, −2.24999537054395255814031732844, −1.69602031983194701587696257715, −1.54281582164676335827024476909, −0.31977648282831013453698484200,
0.31977648282831013453698484200, 1.54281582164676335827024476909, 1.69602031983194701587696257715, 2.24999537054395255814031732844, 2.36355300339078329178547703560, 2.40791628965121693569864741095, 2.92011694089387734672592693340, 3.09871676134091068327887002135, 3.72195605877960860319738002971, 3.72873554234885787896417859087, 4.48684567522907652550270173485, 4.52509322663936736857387353690, 4.72048933824831001552265034074, 4.80169613690520254540019697438, 5.13388686821245546767550560501, 5.60639579179183138131763595348, 5.68566513896079986789412405246, 5.73087250125819981732741227793, 6.10673235977313205134280288605, 6.47814749218052134902352559733, 6.52385759139164101756742543078, 6.85319656408317174363087387904, 7.16419553915918597119040530313, 7.36451737080328065689132144795, 7.43260939826399787141804643430
