| L(s) = 1 | − 5·2-s + 4·3-s + 12·4-s − 5·5-s − 20·6-s + 7-s − 20·8-s + 10·9-s + 25·10-s + 11-s + 48·12-s − 24·13-s − 5·14-s − 20·15-s + 30·16-s + 17-s − 50·18-s − 8·19-s − 60·20-s + 4·21-s − 5·22-s − 11·23-s − 80·24-s + 10·25-s + 120·26-s + 20·27-s + 12·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 2.30·3-s + 6·4-s − 2.23·5-s − 8.16·6-s + 0.377·7-s − 7.07·8-s + 10/3·9-s + 7.90·10-s + 0.301·11-s + 13.8·12-s − 6.65·13-s − 1.33·14-s − 5.16·15-s + 15/2·16-s + 0.242·17-s − 11.7·18-s − 1.83·19-s − 13.4·20-s + 0.872·21-s − 1.06·22-s − 2.29·23-s − 16.3·24-s + 2·25-s + 23.5·26-s + 3.84·27-s + 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 17 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 17 | $C_2^2:C_4$ | \( 1 - T + 14 T^{2} - 37 T^{3} + 359 T^{4} - 37 p T^{5} + 14 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 8 T + 15 T^{2} - 122 T^{3} - 1021 T^{4} - 122 p T^{5} + 15 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 11 T + 38 T^{2} + 125 T^{3} + 821 T^{4} + 125 p T^{5} + 38 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 14 T + 67 T^{2} + 8 p T^{3} + 45 p T^{4} + 8 p^{2} T^{5} + 67 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 13 T + 38 T^{2} - 109 T^{3} - 995 T^{4} - 109 p T^{5} + 38 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 3 T + 17 T^{2} - 225 T^{3} + 2116 T^{4} - 225 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 9 T - 10 T^{2} - 339 T^{3} - 1601 T^{4} - 339 p T^{5} - 10 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_4\times C_2$ | \( 1 - T - 52 T^{2} + 105 T^{3} + 2651 T^{4} + 105 p T^{5} - 52 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 20 T + 101 T^{2} - 390 T^{3} - 6739 T^{4} - 390 p T^{5} + 101 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 5 T - 27 T^{2} + 385 T^{3} + 6524 T^{4} + 385 p T^{5} - 27 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 13 T - 2 T^{2} - 349 T^{3} + 405 T^{4} - 349 p T^{5} - 2 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 4 T - 57 T^{2} + 170 T^{3} + 6221 T^{4} + 170 p T^{5} - 57 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 16 T + 177 T^{2} + 2168 T^{3} + 25505 T^{4} + 2168 p T^{5} + 177 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 24 T + 223 T^{2} + 1830 T^{3} + 18931 T^{4} + 1830 p T^{5} + 223 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 16 T + 257 T^{2} + 2868 T^{3} + 35165 T^{4} + 2868 p T^{5} + 257 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 17 T + 12 T^{2} - 1445 T^{3} - 15649 T^{4} - 1445 p T^{5} + 12 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.74196294594757650655064228510, −7.58763221538314610853346521415, −7.51687072516596226603554143489, −7.37350298384482589209268745048, −7.34798061559415692634722596141, −7.20732095956231384172244018831, −6.94514162048783402392379798476, −6.71296247319353262223706175768, −5.94236397877721248269784948177, −5.83362192333199917742612595730, −5.64390027508103111346320025122, −5.09463934852671097936617451596, −4.66378598874927086404231237800, −4.63652699871767078287635805360, −4.55964022260652443507550775659, −3.97388040456725791509372221938, −3.77854048218075653157474898494, −3.71653058163608680230468772874, −3.22242610827965033039974850356, −2.66480727410812985344973786745, −2.63208539138304102959892375959, −2.35647441768233872859518461119, −2.09901032974110689256623466548, −1.72393696323011989326539785248, −1.54079653817605911296545185860, 0, 0, 0, 0,
1.54079653817605911296545185860, 1.72393696323011989326539785248, 2.09901032974110689256623466548, 2.35647441768233872859518461119, 2.63208539138304102959892375959, 2.66480727410812985344973786745, 3.22242610827965033039974850356, 3.71653058163608680230468772874, 3.77854048218075653157474898494, 3.97388040456725791509372221938, 4.55964022260652443507550775659, 4.63652699871767078287635805360, 4.66378598874927086404231237800, 5.09463934852671097936617451596, 5.64390027508103111346320025122, 5.83362192333199917742612595730, 5.94236397877721248269784948177, 6.71296247319353262223706175768, 6.94514162048783402392379798476, 7.20732095956231384172244018831, 7.34798061559415692634722596141, 7.37350298384482589209268745048, 7.51687072516596226603554143489, 7.58763221538314610853346521415, 7.74196294594757650655064228510
