| L(s) = 1 | + 14·2-s − 81·3-s + 76·4-s + 370·5-s − 1.13e3·6-s − 48·7-s − 152·8-s + 4.37e3·9-s + 5.18e3·10-s − 2.92e3·11-s − 6.15e3·12-s − 672·14-s − 2.99e4·15-s − 2.01e4·16-s − 4.33e4·17-s + 6.12e4·18-s + 5.07e4·19-s + 2.81e4·20-s + 3.88e3·21-s − 4.08e4·22-s − 1.84e4·23-s + 1.23e4·24-s − 3.26e4·25-s − 1.96e5·27-s − 3.64e3·28-s + 1.14e5·29-s − 4.19e5·30-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 1.73·3-s + 0.593·4-s + 1.32·5-s − 2.14·6-s − 0.0528·7-s − 0.104·8-s + 2·9-s + 1.63·10-s − 0.661·11-s − 1.02·12-s − 0.0654·14-s − 2.29·15-s − 1.22·16-s − 2.14·17-s + 2.47·18-s + 1.69·19-s + 0.785·20-s + 0.0916·21-s − 0.818·22-s − 0.315·23-s + 0.181·24-s − 0.418·25-s − 1.92·27-s − 0.0314·28-s + 0.872·29-s − 2.83·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.236377829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.236377829\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{3} T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 7 p T + 15 p^{3} T^{2} - 29 p^{4} T^{3} + 15 p^{10} T^{4} - 7 p^{15} T^{5} + p^{21} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 74 p T + 6783 p^{2} T^{2} - 361412 p^{3} T^{3} + 6783 p^{9} T^{4} - 74 p^{15} T^{5} + p^{21} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 48 T + 606297 T^{2} - 326905648 T^{3} + 606297 p^{7} T^{4} + 48 p^{14} T^{5} + p^{21} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2920 T + 51192921 T^{2} + 114474253936 T^{3} + 51192921 p^{7} T^{4} + 2920 p^{14} T^{5} + p^{21} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 43354 T + 1678811871 T^{2} + 35671407901420 T^{3} + 1678811871 p^{7} T^{4} + 43354 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 50756 T + 3057603829 T^{2} - 91203096420280 T^{3} + 3057603829 p^{7} T^{4} - 50756 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 18408 T + 5834587701 T^{2} - 505805615952 T^{3} + 5834587701 p^{7} T^{4} + 18408 p^{14} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 114554 T + 27091860699 T^{2} - 1444460221648604 T^{3} + 27091860699 p^{7} T^{4} - 114554 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3008 p T + 54305475953 T^{2} + 3915771458447024 T^{3} + 54305475953 p^{7} T^{4} + 3008 p^{15} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 363262 T + 248525531939 T^{2} - 62354386143671764 T^{3} + 248525531939 p^{7} T^{4} - 363262 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 556398 T + 537190691115 T^{2} - 217499379788245500 T^{3} + 537190691115 p^{7} T^{4} - 556398 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 1393420 T + 1255587064921 T^{2} - 734887073996178056 T^{3} + 1255587064921 p^{7} T^{4} - 1393420 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 938604 T + 1059194572125 T^{2} - 903245815050075624 T^{3} + 1059194572125 p^{7} T^{4} - 938604 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 1293270 T + 1126420357443 T^{2} + 348373048259584068 T^{3} + 1126420357443 p^{7} T^{4} + 1293270 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5846928 T + 18201271969785 T^{2} - 35404515146078666592 T^{3} + 18201271969785 p^{7} T^{4} - 5846928 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6300502 T + 22271295833915 T^{2} + 48140518732183166980 T^{3} + 22271295833915 p^{7} T^{4} + 6300502 p^{14} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3619508 T + 13604686566053 T^{2} + 24998151524518222232 T^{3} + 13604686566053 p^{7} T^{4} + 3619508 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 5742612 T + 30399777307173 T^{2} - 96402615964471346328 T^{3} + 30399777307173 p^{7} T^{4} - 5742612 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6208602 T + 38504375209479 T^{2} - \)\(13\!\cdots\!76\)\( T^{3} + 38504375209479 p^{7} T^{4} - 6208602 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 5789976 T + 39397377746829 T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + 39397377746829 p^{7} T^{4} + 5789976 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1217504 T + 59606578734513 T^{2} - 97072643101540657472 T^{3} + 59606578734513 p^{7} T^{4} - 1217504 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16081814 T + 212891431997883 T^{2} - \)\(15\!\cdots\!92\)\( T^{3} + 212891431997883 p^{7} T^{4} - 16081814 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 8275742 T + 234923319509087 T^{2} + \)\(12\!\cdots\!92\)\( T^{3} + 234923319509087 p^{7} T^{4} + 8275742 p^{14} T^{5} + p^{21} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.946795856184620509064939189124, −8.042376553589009234168391513304, −7.85964543754193191389697444273, −7.38206172052960441702119441850, −7.37766969133852452969723878484, −6.75861476407041026533156651845, −6.45629585791420839675702044900, −6.30250561479498346872485181985, −5.95860050365840341968038780986, −5.82543721472079144755368146170, −5.39620404033274685852032808662, −5.10883131901011885627535903959, −4.98596756316902521587368561026, −4.37315475769008038340021359527, −4.26304829317023936435916087645, −4.09135411414424773580324800579, −3.48607254279612970127870573862, −2.89848813998602352533299938218, −2.45948533335157297773066006588, −2.35482643717950899098640765954, −1.96058412758155036721472985420, −1.46499223529458165942034460253, −0.888022508493808784535441904163, −0.68191023092663479079028006231, −0.23722130531567703922662291074,
0.23722130531567703922662291074, 0.68191023092663479079028006231, 0.888022508493808784535441904163, 1.46499223529458165942034460253, 1.96058412758155036721472985420, 2.35482643717950899098640765954, 2.45948533335157297773066006588, 2.89848813998602352533299938218, 3.48607254279612970127870573862, 4.09135411414424773580324800579, 4.26304829317023936435916087645, 4.37315475769008038340021359527, 4.98596756316902521587368561026, 5.10883131901011885627535903959, 5.39620404033274685852032808662, 5.82543721472079144755368146170, 5.95860050365840341968038780986, 6.30250561479498346872485181985, 6.45629585791420839675702044900, 6.75861476407041026533156651845, 7.37766969133852452969723878484, 7.38206172052960441702119441850, 7.85964543754193191389697444273, 8.042376553589009234168391513304, 8.946795856184620509064939189124
