L(s) = 1 | + 3·2-s + 3-s + 7·4-s + 3·6-s + 3·7-s + 15·8-s − 11-s + 7·12-s − 6·13-s + 9·14-s + 30·16-s + 13·17-s + 10·19-s + 3·21-s − 3·22-s − 26·23-s + 15·24-s − 18·26-s + 21·28-s + 13·31-s + 57·32-s − 33-s + 39·34-s − 2·37-s + 30·38-s − 6·39-s − 7·41-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 0.577·3-s + 7/2·4-s + 1.22·6-s + 1.13·7-s + 5.30·8-s − 0.301·11-s + 2.02·12-s − 1.66·13-s + 2.40·14-s + 15/2·16-s + 3.15·17-s + 2.29·19-s + 0.654·21-s − 0.639·22-s − 5.42·23-s + 3.06·24-s − 3.53·26-s + 3.96·28-s + 2.33·31-s + 10.0·32-s − 0.174·33-s + 6.68·34-s − 0.328·37-s + 4.86·38-s − 0.960·39-s − 1.09·41-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
\(L(1)\) |
\(\approx\) |
\(27.68971697\) |
\(L(\frac12)\) |
\(\approx\) |
\(27.68971697\) |
\(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 13 T + 52 T^{2} + 25 T^{3} - 729 T^{4} + 25 p T^{5} + 52 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 10 p T^{3} - 1519 T^{4} + 10 p^{2} T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 19 T^{2} - 120 T^{3} + 721 T^{4} - 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 13 T + 33 T^{2} + 379 T^{3} - 3700 T^{4} + 379 p T^{5} + 33 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 2 T + 27 T^{2} + 160 T^{3} + 1841 T^{4} + 160 p T^{5} + 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 7 T - 22 T^{2} - 161 T^{3} + 575 T^{4} - 161 p T^{5} - 22 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 13 T + 67 T^{2} + 115 T^{3} - 3864 T^{4} + 115 p T^{5} + 67 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 6 T + 23 T^{2} + 480 T^{3} + 5581 T^{4} + 480 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 15 T + 31 T^{2} + 15 p T^{3} - 10424 T^{4} + 15 p^{2} T^{5} + 31 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 3 T + 18 T^{2} + 209 T^{3} + 675 T^{4} + 209 p T^{5} + 18 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 7 T - 2 T^{2} + 569 T^{3} + 8925 T^{4} + 569 p T^{5} - 2 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 4 T - 27 T^{2} - 500 T^{3} + 7301 T^{4} - 500 p T^{5} - 27 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 20 T + 231 T^{2} - 2770 T^{3} + 30671 T^{4} - 2770 p T^{5} + 231 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 9 T + 53 T^{2} + 75 T^{3} - 4004 T^{4} + 75 p T^{5} + 53 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 27 T + 452 T^{2} + 6225 T^{3} + 70951 T^{4} + 6225 p T^{5} + 452 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.40682941293104302028466268793, −7.10843395929492074551079095833, −6.88985406217527221731495307680, −6.53216001747190211997232277448, −6.47763603460499299380985798410, −5.82909025089776788515344878043, −5.75770788991202569900046424278, −5.67878426810135912074319824187, −5.65218694033819430297319202481, −5.09741614886881604890788705933, −4.91049779615965550146332062620, −4.85137707770596154711625846173, −4.65069950605593924050717534739, −4.00459623156620071522734985081, −3.86746099913562477134480236030, −3.82891544337782496695042692717, −3.48110919674696418983579898815, −3.23963773727130479482022917877, −2.62979174371708060695697148657, −2.62489857484836770175719748401, −2.48890871084529067602339968019, −1.86175341525099336484409523059, −1.58118377648209281599832567684, −1.44239997242976584315634446778, −0.791059375388978052098570729558,
0.791059375388978052098570729558, 1.44239997242976584315634446778, 1.58118377648209281599832567684, 1.86175341525099336484409523059, 2.48890871084529067602339968019, 2.62489857484836770175719748401, 2.62979174371708060695697148657, 3.23963773727130479482022917877, 3.48110919674696418983579898815, 3.82891544337782496695042692717, 3.86746099913562477134480236030, 4.00459623156620071522734985081, 4.65069950605593924050717534739, 4.85137707770596154711625846173, 4.91049779615965550146332062620, 5.09741614886881604890788705933, 5.65218694033819430297319202481, 5.67878426810135912074319824187, 5.75770788991202569900046424278, 5.82909025089776788515344878043, 6.47763603460499299380985798410, 6.53216001747190211997232277448, 6.88985406217527221731495307680, 7.10843395929492074551079095833, 7.40682941293104302028466268793