| L(s) = 1 | − 16·2-s + 64·4-s + 48·5-s − 880·7-s + 1.02e3·8-s − 768·10-s − 7.23e3·11-s − 8.56e3·13-s + 1.40e4·14-s − 1.63e4·16-s + 5.14e4·17-s + 7.40e4·19-s + 3.07e3·20-s + 1.15e5·22-s + 5.96e4·23-s − 8.30e4·25-s + 1.36e5·26-s − 5.63e4·28-s + 2.75e5·29-s − 2.45e5·31-s + 6.55e4·32-s − 8.22e5·34-s − 4.22e4·35-s − 1.42e5·37-s − 1.18e6·38-s + 4.91e4·40-s − 4.94e5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.171·5-s − 0.969·7-s + 0.707·8-s − 0.242·10-s − 1.63·11-s − 1.08·13-s + 1.37·14-s − 16-s + 2.53·17-s + 2.47·19-s + 0.0858·20-s + 2.31·22-s + 1.02·23-s − 1.06·25-s + 1.52·26-s − 0.484·28-s + 2.09·29-s − 1.48·31-s + 0.353·32-s − 3.58·34-s − 0.166·35-s − 0.461·37-s − 3.50·38-s + 0.121·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4478594381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4478594381\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 48 T + 85319 T^{2} + 2296944 p T^{3} + 24318984 p^{2} T^{4} + 2296944 p^{8} T^{5} + 85319 p^{14} T^{6} - 48 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 880 T - 826445 T^{2} - 40692080 T^{3} + 1246729941976 T^{4} - 40692080 p^{7} T^{5} - 826445 p^{14} T^{6} + 880 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 7230 T + 15580157 T^{2} - 16495960770 T^{3} - 52043385685092 T^{4} - 16495960770 p^{7} T^{5} + 15580157 p^{14} T^{6} + 7230 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8560 T + 25394566 T^{2} - 664410080000 T^{3} - 6242784224177333 T^{4} - 664410080000 p^{7} T^{5} + 25394566 p^{14} T^{6} + 8560 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 1512 p T + 889914850 T^{2} - 1512 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 37048 T + 955661154 T^{2} - 37048 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 59628 T - 4047090706 T^{2} - 47281318751088 T^{3} + 34100673172237700643 T^{4} - 47281318751088 p^{7} T^{5} - 4047090706 p^{14} T^{6} - 59628 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 275280 T + 35022145082 T^{2} - 1722476703096000 T^{3} + 56389651602085400043 T^{4} - 1722476703096000 p^{7} T^{5} + 35022145082 p^{14} T^{6} - 275280 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 245584 T + 9689482795 T^{2} - 34882513389104 p T^{3} + 152255715491492344 p^{2} T^{4} - 34882513389104 p^{8} T^{5} + 9689482795 p^{14} T^{6} + 245584 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 71060 T + 10424169582 T^{2} + 71060 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 494520 T - 206049016126 T^{2} + 30210473448773280 T^{3} + \)\(11\!\cdots\!15\)\( T^{4} + 30210473448773280 p^{7} T^{5} - 206049016126 p^{14} T^{6} + 494520 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 825280 T - 8568232130 T^{2} + 120505807925428480 T^{3} + \)\(21\!\cdots\!51\)\( T^{4} + 120505807925428480 p^{7} T^{5} - 8568232130 p^{14} T^{6} + 825280 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 927708 T - 198532831378 T^{2} - 42608444421122928 T^{3} + \)\(42\!\cdots\!63\)\( T^{4} - 42608444421122928 p^{7} T^{5} - 198532831378 p^{14} T^{6} - 927708 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 1382112 T + 2812584218785 T^{2} + 1382112 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1588920 T - 2836225583542 T^{2} + 609492838152631680 T^{3} + \)\(16\!\cdots\!03\)\( T^{4} + 609492838152631680 p^{7} T^{5} - 2836225583542 p^{14} T^{6} + 1588920 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 3021968 T + 577547114326 T^{2} - 6857624475464619008 T^{3} + \)\(34\!\cdots\!59\)\( T^{4} - 6857624475464619008 p^{7} T^{5} + 577547114326 p^{14} T^{6} - 3021968 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1882160 T - 8650756632722 T^{2} - 135251429756260160 T^{3} + \)\(90\!\cdots\!55\)\( T^{4} - 135251429756260160 p^{7} T^{5} - 8650756632722 p^{14} T^{6} - 1882160 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3394440 T + 12834752041582 T^{2} + 3394440 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 895090 T + 16713759410403 T^{2} - 895090 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 369920 T - 35661563929118 T^{2} + 965274144557056000 T^{3} + \)\(91\!\cdots\!43\)\( T^{4} + 965274144557056000 p^{7} T^{5} - 35661563929118 p^{14} T^{6} - 369920 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9352050 T + 13841282304221 T^{2} - \)\(18\!\cdots\!50\)\( T^{3} + \)\(23\!\cdots\!12\)\( T^{4} - \)\(18\!\cdots\!50\)\( p^{7} T^{5} + 13841282304221 p^{14} T^{6} - 9352050 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 912960 T + 67864694086222 T^{2} + 912960 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1359790 T - 107845269581135 T^{2} - 70576188445336251890 T^{3} + \)\(53\!\cdots\!56\)\( T^{4} - 70576188445336251890 p^{7} T^{5} - 107845269581135 p^{14} T^{6} + 1359790 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−8.063012380186254513330501689665, −7.87056000998536769566307050566, −7.61923080625313212158563642673, −7.39572134703213001738992273912, −7.05758351844038255347837691193, −6.93014333809001464205309650449, −6.49189588823985485863873120823, −6.03131611559029622807804003709, −5.78930073222790034873443688987, −5.26678086408763781597638371877, −5.15761962463995308946156322854, −5.15258410321028954974597275927, −4.83622531096427063434486509579, −4.12801279415536949339760584165, −3.67903809351569460899624066067, −3.43478499253081713673226127234, −3.07005824133388019483695744360, −2.93547467207926119641724750180, −2.45253809025608908070017329949, −2.18131081035563896072662094304, −1.36296713629566567329000184238, −1.20154424740003537013402286102, −1.12182510581271305340174233032, −0.39368407787198132571757313991, −0.19282368840577195477388045213,
0.19282368840577195477388045213, 0.39368407787198132571757313991, 1.12182510581271305340174233032, 1.20154424740003537013402286102, 1.36296713629566567329000184238, 2.18131081035563896072662094304, 2.45253809025608908070017329949, 2.93547467207926119641724750180, 3.07005824133388019483695744360, 3.43478499253081713673226127234, 3.67903809351569460899624066067, 4.12801279415536949339760584165, 4.83622531096427063434486509579, 5.15258410321028954974597275927, 5.15761962463995308946156322854, 5.26678086408763781597638371877, 5.78930073222790034873443688987, 6.03131611559029622807804003709, 6.49189588823985485863873120823, 6.93014333809001464205309650449, 7.05758351844038255347837691193, 7.39572134703213001738992273912, 7.61923080625313212158563642673, 7.87056000998536769566307050566, 8.063012380186254513330501689665
