| L(s) = 1 | − 16·2-s + 192·4-s + 48·5-s + 880·7-s − 2.04e3·8-s − 768·10-s − 7.23e3·11-s + 8.56e3·13-s − 1.40e4·14-s + 2.04e4·16-s − 2.57e4·17-s + 3.70e4·19-s + 9.21e3·20-s + 1.15e5·22-s + 5.96e4·23-s + 8.53e4·25-s − 1.36e5·26-s + 1.68e5·28-s + 2.75e5·29-s + 2.45e5·31-s − 1.96e5·32-s + 4.11e5·34-s + 4.22e4·35-s − 7.10e4·37-s − 5.92e5·38-s − 9.83e4·40-s − 4.94e5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.171·5-s + 0.969·7-s − 1.41·8-s − 0.242·10-s − 1.63·11-s + 1.08·13-s − 1.37·14-s + 5/4·16-s − 1.26·17-s + 1.23·19-s + 0.257·20-s + 2.31·22-s + 1.02·23-s + 1.09·25-s − 1.52·26-s + 1.45·28-s + 2.09·29-s + 1.48·31-s − 1.06·32-s + 1.79·34-s + 0.166·35-s − 0.230·37-s − 1.75·38-s − 0.242·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.570880575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570880575\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 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 | $D_{4}$ | \( 1 - 1359790 T + 109694298425235 T^{2} - 1359790 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.00550826413757011656437013181, −13.70539814635622610007197247322, −13.01686871374724110544723872619, −12.21337007889613733244470507729, −11.63694195640972417054122918633, −10.99301593278893985667580095263, −10.44061861683896651232177154622, −10.38705883796489373220156977554, −9.084217536575638356132525862329, −8.912200062978743042073135297060, −8.052058287868870660656256556123, −7.82172022711895703202453878806, −6.86740943468252261131551737018, −6.28667707813411758072371437567, −5.21755489106430345248743566775, −4.65559584462011861030171056146, −3.02945006661401698851456633458, −2.47574378774929492262804381124, −1.28793390538577203914539181580, −0.68800615456633406890434157671,
0.68800615456633406890434157671, 1.28793390538577203914539181580, 2.47574378774929492262804381124, 3.02945006661401698851456633458, 4.65559584462011861030171056146, 5.21755489106430345248743566775, 6.28667707813411758072371437567, 6.86740943468252261131551737018, 7.82172022711895703202453878806, 8.052058287868870660656256556123, 8.912200062978743042073135297060, 9.084217536575638356132525862329, 10.38705883796489373220156977554, 10.44061861683896651232177154622, 10.99301593278893985667580095263, 11.63694195640972417054122918633, 12.21337007889613733244470507729, 13.01686871374724110544723872619, 13.70539814635622610007197247322, 14.00550826413757011656437013181
