| L(s) = 1 | + 21·2-s − 84·3-s + 231·4-s − 1.55e3·5-s − 1.76e3·6-s + 2.86e3·8-s − 3.89e4·9-s − 3.26e4·10-s − 3.44e3·11-s − 1.94e4·12-s + 1.97e4·13-s + 1.30e5·15-s − 4.49e4·16-s − 1.01e6·17-s − 8.18e5·18-s − 2.22e5·19-s − 3.58e5·20-s − 7.23e4·22-s + 1.88e6·23-s − 2.40e5·24-s − 1.85e5·25-s + 4.15e5·26-s + 3.84e6·27-s + 4.08e6·29-s + 2.74e6·30-s − 2.86e6·31-s + 2.26e6·32-s + ⋯ |
| L(s) = 1 | + 0.928·2-s − 0.598·3-s + 0.451·4-s − 1.11·5-s − 0.555·6-s + 0.247·8-s − 1.98·9-s − 1.03·10-s − 0.0709·11-s − 0.270·12-s + 0.192·13-s + 0.665·15-s − 0.171·16-s − 2.95·17-s − 1.83·18-s − 0.392·19-s − 0.501·20-s − 0.0658·22-s + 1.40·23-s − 0.148·24-s − 0.0950·25-s + 0.178·26-s + 1.39·27-s + 1.07·29-s + 0.617·30-s − 0.558·31-s + 0.381·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117649 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117649 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7877770746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7877770746\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 21 T + 105 p T^{2} - 303 p^{3} T^{3} + 105 p^{10} T^{4} - 21 p^{18} T^{5} + p^{27} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 28 p T + 5117 p^{2} T^{2} + 122360 p^{3} T^{3} + 5117 p^{11} T^{4} + 28 p^{19} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 1554 T + 520107 p T^{2} + 46663764 p^{2} T^{3} + 520107 p^{10} T^{4} + 1554 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3444 T + 455343105 T^{2} + 125101303155960 T^{3} + 455343105 p^{9} T^{4} + 3444 p^{18} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 19782 T + 18882215055 T^{2} - 378008000651932 T^{3} + 18882215055 p^{9} T^{4} - 19782 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 1016694 T + 649258140783 T^{2} + 263109059935862868 T^{3} + 649258140783 p^{9} T^{4} + 1016694 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 222852 T + 614081373717 T^{2} + 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} + 222852 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 81984 p T + 5417652680517 T^{2} - 5817973976209389696 T^{3} + 5417652680517 p^{9} T^{4} - 81984 p^{19} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4081818 T + 38739015783987 T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + 38739015783987 p^{9} T^{4} - 4081818 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2869440 T + 21142500166221 T^{2} + \)\(22\!\cdots\!64\)\( T^{3} + 21142500166221 p^{9} T^{4} + 2869440 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 1395618 T + 262675972194027 T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + 262675972194027 p^{9} T^{4} - 1395618 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14420658 T + 764979654799959 T^{2} - \)\(74\!\cdots\!64\)\( T^{3} + 764979654799959 p^{9} T^{4} - 14420658 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 61631172 T + 2687603003165025 T^{2} + \)\(16\!\cdots\!04\)\( p T^{3} + 2687603003165025 p^{9} T^{4} + 61631172 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10368960 T + 2946826961339709 T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + 2946826961339709 p^{9} T^{4} - 10368960 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 67502610 T + 6295046710287531 T^{2} - \)\(20\!\cdots\!32\)\( T^{3} + 6295046710287531 p^{9} T^{4} - 67502610 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 42590100 T + 19076976504365997 T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + 19076976504365997 p^{9} T^{4} - 42590100 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 191746842 T + 38596678668907359 T^{2} + \)\(44\!\cdots\!36\)\( T^{3} + 38596678668907359 p^{9} T^{4} + 191746842 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 255175788 T + 80172518705654361 T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + 80172518705654361 p^{9} T^{4} + 255175788 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 296514504 T + 147895725194380437 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + 147895725194380437 p^{9} T^{4} - 296514504 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 344213310 T + 83298340302082311 T^{2} + \)\(20\!\cdots\!12\)\( T^{3} + 83298340302082311 p^{9} T^{4} + 344213310 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 960412656 T + 566786434394061357 T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + 566786434394061357 p^{9} T^{4} + 960412656 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1100517180 T + 873896700882341301 T^{2} - \)\(43\!\cdots\!28\)\( T^{3} + 873896700882341301 p^{9} T^{4} - 1100517180 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 506816478 T + 956194525794688887 T^{2} + \)\(35\!\cdots\!64\)\( T^{3} + 956194525794688887 p^{9} T^{4} + 506816478 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 647498250 T + 819696244591424799 T^{2} - \)\(48\!\cdots\!84\)\( T^{3} + 819696244591424799 p^{9} T^{4} - 647498250 p^{18} T^{5} + p^{27} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93826461536541798916442884718, −11.54264747497163780752124895529, −11.48337130194418745275063140926, −11.21980406440912404157383541710, −10.63591792291448371981357466465, −10.51975832955251053524617297718, −9.652403063960781176287531495382, −8.788889603358964408111648919238, −8.759747873440187258200565025844, −8.638105570208761576369299348314, −7.87830764494977187241774138755, −7.33448568532706084735079870740, −6.74161111924769806652191575934, −6.49272240549395913862677195741, −5.96639948110595423697251286766, −5.54381466530330578979508234755, −4.76373081616311378306392195466, −4.61715652542764830031876041140, −4.20036744200007227548791391368, −3.30163654169432214955835818829, −3.04013122834108640926785942800, −2.40021280111603754653199059999, −1.82067337280000974999082691148, −0.62719757498885462360226772020, −0.24738704476899092627356511163,
0.24738704476899092627356511163, 0.62719757498885462360226772020, 1.82067337280000974999082691148, 2.40021280111603754653199059999, 3.04013122834108640926785942800, 3.30163654169432214955835818829, 4.20036744200007227548791391368, 4.61715652542764830031876041140, 4.76373081616311378306392195466, 5.54381466530330578979508234755, 5.96639948110595423697251286766, 6.49272240549395913862677195741, 6.74161111924769806652191575934, 7.33448568532706084735079870740, 7.87830764494977187241774138755, 8.638105570208761576369299348314, 8.759747873440187258200565025844, 8.788889603358964408111648919238, 9.652403063960781176287531495382, 10.51975832955251053524617297718, 10.63591792291448371981357466465, 11.21980406440912404157383541710, 11.48337130194418745275063140926, 11.54264747497163780752124895529, 11.93826461536541798916442884718
