| L(s) = 1 | + 33·2-s + 89·3-s − 53·4-s + 2.93e3·6-s + 5.25e3·7-s − 1.93e4·8-s + 3.55e3·9-s − 5.46e4·11-s − 4.71e3·12-s + 2.15e5·13-s + 1.73e5·14-s − 5.02e5·16-s + 3.34e5·17-s + 1.17e5·18-s + 8.18e5·19-s + 4.67e5·21-s − 1.80e6·22-s + 3.52e6·23-s − 1.72e6·24-s + 7.12e6·26-s + 1.70e6·27-s − 2.78e5·28-s + 2.17e6·29-s + 4.27e6·31-s − 6.15e6·32-s − 4.86e6·33-s + 1.10e7·34-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.634·3-s − 0.103·4-s + 0.925·6-s + 0.827·7-s − 1.67·8-s + 0.180·9-s − 1.12·11-s − 0.0656·12-s + 2.09·13-s + 1.20·14-s − 1.91·16-s + 0.972·17-s + 0.263·18-s + 1.44·19-s + 0.525·21-s − 1.64·22-s + 2.62·23-s − 1.06·24-s + 3.05·26-s + 0.618·27-s − 0.0856·28-s + 0.571·29-s + 0.831·31-s − 1.03·32-s − 0.714·33-s + 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(10.08344231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.08344231\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 33 T + 571 p T^{2} - 2505 p^{3} T^{3} + 571 p^{10} T^{4} - 33 p^{18} T^{5} + p^{27} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 89 T + 1456 p T^{2} - 197885 p^{2} T^{3} + 1456 p^{10} T^{4} - 89 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5258 T + 11082951 p T^{2} - 9176976100 p^{2} T^{3} + 11082951 p^{10} T^{4} - 5258 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 5816187440 p^{9} T^{4} + 54699 p^{18} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 215884 T + 31225156343 T^{2} - 3094983572287480 T^{3} + 31225156343 p^{9} T^{4} - 215884 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 334983 T + 355030662062 T^{2} - 77675997792847395 T^{3} + 355030662062 p^{9} T^{4} - 334983 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 818845 T + 468173656712 T^{2} - 302339390932836385 T^{3} + 468173656712 p^{9} T^{4} - 818845 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3526854 T + 8700596724993 T^{2} - 13086552252942250620 T^{3} + 8700596724993 p^{9} T^{4} - 3526854 p^{18} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2175480 T + 1119231270883 p T^{2} - 59155485309271560240 T^{3} + 1119231270883 p^{10} T^{4} - 2175480 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4274066 T + 82851493809465 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + 82851493809465 p^{9} T^{4} - 4274066 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10305042 T + 340842196290747 T^{2} + \)\(25\!\cdots\!40\)\( T^{3} + 340842196290747 p^{9} T^{4} + 10305042 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 5926311 T + 232043727124790 T^{2} - \)\(23\!\cdots\!95\)\( T^{3} + 232043727124790 p^{9} T^{4} - 5926311 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 24429956 T + 1676050268074193 T^{2} + \)\(57\!\cdots\!00\)\( p T^{3} + 1676050268074193 p^{9} T^{4} + 24429956 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 66858708 T + 4517266130065277 T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + 4517266130065277 p^{9} T^{4} - 66858708 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 132620514 T + 9209248487675243 T^{2} - \)\(47\!\cdots\!40\)\( T^{3} + 9209248487675243 p^{9} T^{4} - 132620514 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5670960 T + 19337695182532817 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + 19337695182532817 p^{9} T^{4} - 5670960 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 125306926 T + 22192130203358915 T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + 22192130203358915 p^{9} T^{4} - 125306926 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 88829483 T + 30562384919894712 T^{2} - \)\(76\!\cdots\!95\)\( T^{3} + 30562384919894712 p^{9} T^{4} - 88829483 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 297550596 T + 87036096018332165 T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + 87036096018332165 p^{9} T^{4} - 297550596 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 181321729 T + 94365252526618118 T^{2} - \)\(75\!\cdots\!45\)\( T^{3} + 94365252526618118 p^{9} T^{4} - 181321729 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 310025170 T + 176092553119892457 T^{2} + \)\(48\!\cdots\!60\)\( T^{3} + 176092553119892457 p^{9} T^{4} + 310025170 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 731088801 T + 685433348047337568 T^{2} + \)\(27\!\cdots\!65\)\( T^{3} + 685433348047337568 p^{9} T^{4} + 731088801 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1103860035 T + 1320664213408319502 T^{2} + \)\(75\!\cdots\!55\)\( T^{3} + 1320664213408319502 p^{9} T^{4} + 1103860035 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 332236842 T + 1487745026246658927 T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + 1487745026246658927 p^{9} T^{4} + 332236842 p^{18} T^{5} + p^{27} T^{6} \) |
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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
−14.05377358745202535926644625964, −13.37969233562154408952193066941, −13.11958016445962454334005890977, −12.81826532078327609091150980471, −12.32705528020000392422126122940, −11.58180805437085056194649124014, −11.26786933341392124843049168507, −10.94024671919482620157877047436, −10.03361091542299569749572357312, −9.908251700743041153505721717250, −8.791045451705693979459405422968, −8.650874549281987437354782120637, −8.575300177523225743878026733772, −7.59608868717520233109273313875, −7.13789794216192565082366786241, −6.40392045672942535949071900421, −5.50833581116018893175617412934, −5.26764260880167384616255571225, −4.85804658318435035742082982376, −4.17952865973371480268657006001, −3.39096790567837368061594916585, −3.21764120258494620253422648474, −2.37442858380724106393917883437, −1.07674648065386974577764827995, −0.837243027232633801189288092848,
0.837243027232633801189288092848, 1.07674648065386974577764827995, 2.37442858380724106393917883437, 3.21764120258494620253422648474, 3.39096790567837368061594916585, 4.17952865973371480268657006001, 4.85804658318435035742082982376, 5.26764260880167384616255571225, 5.50833581116018893175617412934, 6.40392045672942535949071900421, 7.13789794216192565082366786241, 7.59608868717520233109273313875, 8.575300177523225743878026733772, 8.650874549281987437354782120637, 8.791045451705693979459405422968, 9.908251700743041153505721717250, 10.03361091542299569749572357312, 10.94024671919482620157877047436, 11.26786933341392124843049168507, 11.58180805437085056194649124014, 12.32705528020000392422126122940, 12.81826532078327609091150980471, 13.11958016445962454334005890977, 13.37969233562154408952193066941, 14.05377358745202535926644625964
