| L(s) = 1 | + 48·5-s − 880·7-s + 7.23e3·11-s + 8.56e3·13-s − 2.57e4·17-s − 3.70e4·19-s − 5.96e4·23-s + 8.53e4·25-s + 2.75e5·29-s − 2.45e5·31-s − 4.22e4·35-s − 7.10e4·37-s − 4.94e5·41-s − 8.25e5·43-s − 9.27e5·47-s − 8.26e5·49-s + 1.38e6·53-s + 3.47e5·55-s + 1.58e6·59-s − 3.02e6·61-s + 4.10e5·65-s + 1.88e6·67-s − 3.39e6·71-s + 8.95e5·73-s − 6.36e6·77-s + 3.69e5·79-s − 9.35e6·83-s + ⋯ |
| L(s) = 1 | + 0.171·5-s − 0.969·7-s + 1.63·11-s + 1.08·13-s − 1.26·17-s − 1.23·19-s − 1.02·23-s + 1.09·25-s + 2.09·29-s − 1.48·31-s − 0.166·35-s − 0.230·37-s − 1.12·41-s − 1.58·43-s − 1.30·47-s − 1.00·49-s + 1.27·53-s + 0.281·55-s + 1.00·59-s − 1.70·61-s + 0.185·65-s + 0.764·67-s − 1.12·71-s + 0.269·73-s − 1.58·77-s + 0.0844·79-s − 1.79·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8151438176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8151438176\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 48 T - 16603 p T^{2} - 48 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 880 T + 1600845 T^{2} + 880 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7230 T + 36692743 T^{2} - 7230 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8560 T + 47879034 T^{2} - 8560 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1512 p T + 889914850 T^{2} + 1512 p^{8} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 37048 T + 955661154 T^{2} + 37048 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 59628 T + 7602589090 T^{2} + 59628 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 275280 T + 40756933318 T^{2} - 275280 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 245584 T + 1632968331 p T^{2} + 245584 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 71060 T + 10424169582 T^{2} + 71060 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 494520 T + 450599046526 T^{2} + 494520 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 825280 T + 689655310530 T^{2} + 825280 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 927708 T + 1059174964642 T^{2} + 927708 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1382112 T + 2812584218785 T^{2} - 1382112 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1588920 T + 5360892349942 T^{2} - 1588920 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3021968 T + 8554743478698 T^{2} + 3021968 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1882160 T + 12193282898322 T^{2} - 1882160 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3394440 T + 12834752041582 T^{2} + 3394440 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 895090 T + 16713759410403 T^{2} - 895090 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 369920 T + 35798404735518 T^{2} - 369920 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9352050 T + 73619556898279 T^{2} + 9352050 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 912960 T + 67864694086222 T^{2} - 912960 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1359790 T + 109694298425235 T^{2} - 1359790 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.17690876694369302758743140034, −9.725756102121372502689710929111, −9.341649057527461042489188766044, −8.692426483875793388892762902807, −8.490652512409446085272111621306, −8.327161465668679816058320208146, −7.20499882777946582195971764911, −6.67164516132621093271841426177, −6.54149176835181672433743047569, −6.33393939559507683310903071207, −5.61093198974936901330453708463, −4.90257079123829652601511741586, −4.24108583561473711662001392672, −4.05315003837202276378205350561, −3.22863942259602460676353496576, −3.04478889961958693300413877686, −1.92918410052246784114236775789, −1.72359026825984855110007941201, −0.995270845335603896618396769169, −0.19481299066556087932263222449,
0.19481299066556087932263222449, 0.995270845335603896618396769169, 1.72359026825984855110007941201, 1.92918410052246784114236775789, 3.04478889961958693300413877686, 3.22863942259602460676353496576, 4.05315003837202276378205350561, 4.24108583561473711662001392672, 4.90257079123829652601511741586, 5.61093198974936901330453708463, 6.33393939559507683310903071207, 6.54149176835181672433743047569, 6.67164516132621093271841426177, 7.20499882777946582195971764911, 8.327161465668679816058320208146, 8.490652512409446085272111621306, 8.692426483875793388892762902807, 9.341649057527461042489188766044, 9.725756102121372502689710929111, 10.17690876694369302758743140034
