| L(s) = 1 | + 2·3-s − 4-s − 4·5-s + 2·9-s + 2·11-s − 2·12-s − 4·13-s − 8·15-s + 4·16-s − 2·17-s + 4·19-s + 4·20-s + 4·23-s + 14·25-s − 8·27-s + 16·29-s − 10·31-s + 4·33-s − 2·36-s + 8·37-s − 8·39-s + 36·41-s + 32·43-s − 2·44-s − 8·45-s + 10·47-s + 8·48-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 1.78·5-s + 2/3·9-s + 0.603·11-s − 0.577·12-s − 1.10·13-s − 2.06·15-s + 16-s − 0.485·17-s + 0.917·19-s + 0.894·20-s + 0.834·23-s + 14/5·25-s − 1.53·27-s + 2.97·29-s − 1.79·31-s + 0.696·33-s − 1/3·36-s + 1.31·37-s − 1.28·39-s + 5.62·41-s + 4.87·43-s − 0.301·44-s − 1.19·45-s + 1.45·47-s + 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582010080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582010080\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 26 T^{2} - 8 T^{3} + 543 T^{4} - 8 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 6 T^{2} + 64 T^{3} - 181 T^{4} + 64 p T^{5} - 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 14 T^{2} + 64 T^{3} + 3 T^{4} + 64 p T^{5} - 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 10 T + 18 T^{2} + 200 T^{3} + 2663 T^{4} + 200 p T^{5} + 18 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T - 6 T^{2} + 32 T^{3} + 1163 T^{4} + 32 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 10 T - 14 T^{2} - 200 T^{3} + 6087 T^{4} - 200 p T^{5} - 14 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 8 T - 38 T^{2} - 32 T^{3} + 4203 T^{4} - 32 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T - 110 T^{2} + 8 T^{3} + 9279 T^{4} + 8 p T^{5} - 110 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 10 T - 42 T^{2} + 200 T^{3} + 10343 T^{4} + 200 p T^{5} - 42 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 20 T + 186 T^{2} + 1600 T^{3} + 14507 T^{4} + 1600 p T^{5} + 186 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 6 T - 114 T^{2} + 24 T^{3} + 14543 T^{4} + 24 p T^{5} - 114 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 78 T^{2} - 157 T^{4} - 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.71036786314683447360842044892, −7.55704003438682787183079477920, −7.45280500634263382231823822859, −7.39968681691492480206979434580, −7.30483261970378141121782912935, −6.65602461053806146121869447091, −6.37211565424609510374812786034, −6.00933081208038237373277553744, −5.76765834664645120392100170624, −5.71281609949171338462861166550, −5.50182381850062081084515544476, −4.69977730771955953903722951491, −4.52921055138864951451262564194, −4.52816663494985412323786277799, −4.20184981005406760522424458129, −4.00674898605575350253888151167, −3.93020745287735543370333328026, −3.16219838229870067734012668244, −3.02569756970455851140797070376, −2.78795135865811864523611655648, −2.61154015120235771882239822792, −2.28753809640506010224077405796, −1.24448746296506385756608264342, −1.14239053234515431091069104261, −0.58818803448977961567802267588,
0.58818803448977961567802267588, 1.14239053234515431091069104261, 1.24448746296506385756608264342, 2.28753809640506010224077405796, 2.61154015120235771882239822792, 2.78795135865811864523611655648, 3.02569756970455851140797070376, 3.16219838229870067734012668244, 3.93020745287735543370333328026, 4.00674898605575350253888151167, 4.20184981005406760522424458129, 4.52816663494985412323786277799, 4.52921055138864951451262564194, 4.69977730771955953903722951491, 5.50182381850062081084515544476, 5.71281609949171338462861166550, 5.76765834664645120392100170624, 6.00933081208038237373277553744, 6.37211565424609510374812786034, 6.65602461053806146121869447091, 7.30483261970378141121782912935, 7.39968681691492480206979434580, 7.45280500634263382231823822859, 7.55704003438682787183079477920, 7.71036786314683447360842044892
