| L(s) = 1 | + 2-s − 86·3-s − 79·4-s − 198·5-s − 86·6-s − 1.55e3·7-s − 1.59e3·8-s + 967·9-s − 198·10-s + 5.54e3·11-s + 6.79e3·12-s − 1.50e4·13-s − 1.55e3·14-s + 1.70e4·15-s − 6.95e3·16-s + 1.47e4·17-s + 967·18-s − 7.48e3·19-s + 1.56e4·20-s + 1.33e5·21-s + 5.54e3·22-s − 1.94e5·23-s + 1.37e5·24-s − 1.38e5·25-s − 1.50e4·26-s + 1.07e5·27-s + 1.23e5·28-s + ⋯ |
| L(s) = 1 | + 0.0883·2-s − 1.83·3-s − 0.617·4-s − 0.708·5-s − 0.162·6-s − 1.71·7-s − 1.10·8-s + 0.442·9-s − 0.0626·10-s + 1.25·11-s + 1.13·12-s − 1.89·13-s − 0.151·14-s + 1.30·15-s − 0.424·16-s + 0.727·17-s + 0.0390·18-s − 0.250·19-s + 0.437·20-s + 3.15·21-s + 0.110·22-s − 3.33·23-s + 2.02·24-s − 1.77·25-s − 0.167·26-s + 1.04·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$ | \( ( 1 - p^{3} T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 5 p^{4} T^{2} + 359 p^{2} T^{3} + 5 p^{11} T^{4} - p^{14} T^{5} + p^{21} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 86 T + 2143 p T^{2} + 40292 p^{2} T^{3} + 2143 p^{8} T^{4} + 86 p^{14} T^{5} + p^{21} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 198 T + 177571 T^{2} + 31262532 T^{3} + 177571 p^{7} T^{4} + 198 p^{14} T^{5} + p^{21} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 1558 T + 2346921 T^{2} + 2034323780 T^{3} + 2346921 p^{7} T^{4} + 1558 p^{14} T^{5} + p^{21} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5542 T + 57697157 T^{2} - 214597222276 T^{3} + 57697157 p^{7} T^{4} - 5542 p^{14} T^{5} + p^{21} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 15050 T + 10546023 p T^{2} + 918670965468 T^{3} + 10546023 p^{8} T^{4} + 15050 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 7480 T + 1302721905 T^{2} - 7827185642992 T^{3} + 1302721905 p^{7} T^{4} + 7480 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 194838 T + 978095087 p T^{2} + 1578686759869956 T^{3} + 978095087 p^{8} T^{4} + 194838 p^{14} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 225486 T + 12849146875 T^{2} - 1027649768743788 T^{3} + 12849146875 p^{7} T^{4} + 225486 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 197310 T + 29970378577 T^{2} - 3455506190298868 T^{3} + 29970378577 p^{7} T^{4} - 197310 p^{14} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 859374 T + 505394549635 T^{2} + 179028252673028276 T^{3} + 505394549635 p^{7} T^{4} + 859374 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 769806 T + 780191092279 T^{2} - 316324014016920036 T^{3} + 780191092279 p^{7} T^{4} - 769806 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 1018856 T + 983875775817 T^{2} - 555398153170342192 T^{3} + 983875775817 p^{7} T^{4} - 1018856 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 1430440 T + 1796808113837 T^{2} + 1298122931022223792 T^{3} + 1796808113837 p^{7} T^{4} + 1430440 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 788122 T + 3670919588291 T^{2} - 1859101340422767196 T^{3} + 3670919588291 p^{7} T^{4} - 788122 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1371096 T + 6441009631129 T^{2} - 6216633246850251216 T^{3} + 6441009631129 p^{7} T^{4} - 1371096 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 9666 p T + 9478901919451 T^{2} - 3707040753816033436 T^{3} + 9478901919451 p^{7} T^{4} - 9666 p^{15} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4851452 T + 23434724383889 T^{2} + 59980455854394377960 T^{3} + 23434724383889 p^{7} T^{4} + 4851452 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6699398 T + 38387989298969 T^{2} - 1718221084087484092 p T^{3} + 38387989298969 p^{7} T^{4} - 6699398 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 444438 T + 4415357786743 T^{2} - 61063221549199602964 T^{3} + 4415357786743 p^{7} T^{4} - 444438 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1039946 T + 21476622148593 T^{2} - 23830879537077150652 T^{3} + 21476622148593 p^{7} T^{4} - 1039946 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 909784 T + 28508520590609 T^{2} + 30234057398909173808 T^{3} + 28508520590609 p^{7} T^{4} - 909784 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6024450 T + 95135983014343 T^{2} - \)\(51\!\cdots\!56\)\( T^{3} + 95135983014343 p^{7} T^{4} - 6024450 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 12851130 T + 250793138509039 T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + 250793138509039 p^{7} T^{4} + 12851130 p^{14} T^{5} + p^{21} T^{6} \) |
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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
−16.33855596500571609014877518363, −15.81616664124244860195568242032, −15.58131638530431764046869546112, −14.87587004648342920000702239887, −14.23780008166410186057681974864, −14.12221511962331442422597898445, −13.41028586059332584688700855336, −12.58677422870496946604325744172, −12.19200304462970940547563253474, −12.09630137749720379607075869746, −11.59249119515581213497354888841, −11.49298209407449914192027621251, −10.34274193988826722078572201075, −9.938851743426857166263055557181, −9.406828130425363071200918491565, −9.248576255534712006541144424009, −8.082659393969022750044253943520, −7.64057276254912638834149376569, −6.46831418431637528212175736483, −6.35773852132219868048706871019, −5.74417067064533309013165380085, −5.26827037637786956570864017193, −3.97151312972195662597048118509, −3.74803945821057872818799407409, −2.38832777191801862085459766638, 0, 0, 0,
2.38832777191801862085459766638, 3.74803945821057872818799407409, 3.97151312972195662597048118509, 5.26827037637786956570864017193, 5.74417067064533309013165380085, 6.35773852132219868048706871019, 6.46831418431637528212175736483, 7.64057276254912638834149376569, 8.082659393969022750044253943520, 9.248576255534712006541144424009, 9.406828130425363071200918491565, 9.938851743426857166263055557181, 10.34274193988826722078572201075, 11.49298209407449914192027621251, 11.59249119515581213497354888841, 12.09630137749720379607075869746, 12.19200304462970940547563253474, 12.58677422870496946604325744172, 13.41028586059332584688700855336, 14.12221511962331442422597898445, 14.23780008166410186057681974864, 14.87587004648342920000702239887, 15.58131638530431764046869546112, 15.81616664124244860195568242032, 16.33855596500571609014877518363
