| L(s) = 1 | − 2·3-s + 8·5-s + 4·7-s − 9-s + 8·11-s + 8·13-s − 16·15-s − 10·17-s − 8·21-s − 14·23-s + 38·25-s + 2·27-s + 18·29-s + 16·31-s − 16·33-s + 32·35-s − 4·37-s − 16·39-s − 4·41-s − 26·43-s − 8·45-s − 16·47-s + 15·49-s + 20·51-s − 20·53-s + 64·55-s − 24·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3.57·5-s + 1.51·7-s − 1/3·9-s + 2.41·11-s + 2.21·13-s − 4.13·15-s − 2.42·17-s − 1.74·21-s − 2.91·23-s + 38/5·25-s + 0.384·27-s + 3.34·29-s + 2.87·31-s − 2.78·33-s + 5.40·35-s − 0.657·37-s − 2.56·39-s − 0.624·41-s − 3.96·43-s − 1.19·45-s − 2.33·47-s + 15/7·49-s + 2.80·51-s − 2.74·53-s + 8.62·55-s − 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.446971110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.446971110\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + T^{2} - 4 T^{3} + 64 T^{4} - 4 p T^{5} + p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 8 p T^{3} - 52 p T^{4} + 8 p^{2} T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 70 T^{3} + 16 T^{4} + 70 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 96 T^{3} - 124 T^{4} - 96 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 226 p T^{5} + 53 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 18 T + 177 T^{2} - 1242 T^{3} + 7052 T^{4} - 1242 p T^{5} + 177 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 912 T^{3} + 5822 T^{4} - 912 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 59 T^{2} + 4 T^{3} + 3664 T^{4} + 4 p T^{5} - 59 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 424 T^{3} + 2092 T^{4} + 424 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 26 T + 365 T^{2} + 3474 T^{3} + 25520 T^{4} + 3474 p T^{5} + 365 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1580 T^{3} + 11806 T^{4} + 1580 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 153 T^{2} - 1176 T^{3} - 21292 T^{4} - 1176 p T^{5} + 153 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 780 T^{3} + 2952 T^{4} - 780 p T^{5} + 113 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 272 T^{3} - 1748 T^{4} - 272 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 16134 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 28 T + 197 T^{2} - 2408 T^{3} - 48788 T^{4} - 2408 p T^{5} + 197 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T + 253 T^{2} - 2460 T^{3} + 37272 T^{4} - 2460 p T^{5} + 253 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.591950980239454467586998966182, −8.444660537791265423144972916360, −8.394313249203549423575060590861, −8.183443818970159487408686624807, −8.161351086868398573077112588103, −7.07951996508441562862395869197, −6.72453033726633660505346731879, −6.61005388400809259489562530219, −6.54454765080026417625230205035, −6.20638442546027418265760043616, −6.09366342440689977196556217738, −5.93835220952622605392422577653, −5.90594786496542345577679402283, −4.99041875014808462142456253639, −4.89362049532440699378101056381, −4.77956201674702098152430519378, −4.60535029183819381395175427330, −4.10798740537627876937749113685, −3.45780390356201486857207431742, −3.21055211573303064445097567141, −2.62721635842540548966459577979, −2.01387486736479624708741480889, −1.85028993952903655405457105834, −1.38225971131092144365299955548, −1.33051755301066278508759502655,
1.33051755301066278508759502655, 1.38225971131092144365299955548, 1.85028993952903655405457105834, 2.01387486736479624708741480889, 2.62721635842540548966459577979, 3.21055211573303064445097567141, 3.45780390356201486857207431742, 4.10798740537627876937749113685, 4.60535029183819381395175427330, 4.77956201674702098152430519378, 4.89362049532440699378101056381, 4.99041875014808462142456253639, 5.90594786496542345577679402283, 5.93835220952622605392422577653, 6.09366342440689977196556217738, 6.20638442546027418265760043616, 6.54454765080026417625230205035, 6.61005388400809259489562530219, 6.72453033726633660505346731879, 7.07951996508441562862395869197, 8.161351086868398573077112588103, 8.183443818970159487408686624807, 8.394313249203549423575060590861, 8.444660537791265423144972916360, 8.591950980239454467586998966182
