| L(s) = 1 | − 2·3-s + 21·7-s − 10·9-s − 22·11-s − 80·13-s − 88·17-s + 100·19-s − 42·21-s − 136·23-s − 40·27-s + 132·29-s − 108·31-s + 44·33-s − 438·37-s + 160·39-s + 374·41-s − 140·43-s − 230·47-s + 294·49-s + 176·51-s − 238·53-s − 200·57-s − 356·59-s + 1.21e3·61-s − 210·63-s − 636·67-s + 272·69-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.13·7-s − 0.370·9-s − 0.603·11-s − 1.70·13-s − 1.25·17-s + 1.20·19-s − 0.436·21-s − 1.23·23-s − 0.285·27-s + 0.845·29-s − 0.625·31-s + 0.232·33-s − 1.94·37-s + 0.656·39-s + 1.42·41-s − 0.496·43-s − 0.713·47-s + 6/7·49-s + 0.483·51-s − 0.616·53-s − 0.464·57-s − 0.785·59-s + 2.53·61-s − 0.419·63-s − 1.15·67-s + 0.474·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 14 T^{2} + 88 T^{3} + 14 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 p T + 2314 T^{2} + 39520 T^{3} + 2314 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 80 T + 8656 T^{2} + 368686 T^{3} + 8656 p^{3} T^{4} + 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 88 T + 11076 T^{2} + 873274 T^{3} + 11076 p^{3} T^{4} + 88 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 100 T + 879 p T^{2} - 940136 T^{3} + 879 p^{4} T^{4} - 100 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 136 T + 3873 T^{2} - 1243376 T^{3} + 3873 p^{3} T^{4} + 136 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 132 T + 73188 T^{2} - 6102222 T^{3} + 73188 p^{3} T^{4} - 132 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 108 T + 64749 T^{2} + 6669864 T^{3} + 64749 p^{3} T^{4} + 108 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 438 T + 136707 T^{2} + 30296964 T^{3} + 136707 p^{3} T^{4} + 438 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 374 T + 214835 T^{2} - 45912684 T^{3} + 214835 p^{3} T^{4} - 374 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 140 T + 158901 T^{2} + 11143672 T^{3} + 158901 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 230 T + 102690 T^{2} + 7000628 T^{3} + 102690 p^{3} T^{4} + 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 238 T + 161799 T^{2} + 8166316 T^{3} + 161799 p^{3} T^{4} + 238 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 356 T + 384121 T^{2} + 154836248 T^{3} + 384121 p^{3} T^{4} + 356 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1210 T + 1136983 T^{2} - 600859364 T^{3} + 1136983 p^{3} T^{4} - 1210 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 636 T + 325521 T^{2} + 120423464 T^{3} + 325521 p^{3} T^{4} + 636 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 872 T + 230149 T^{2} + 14031536 T^{3} + 230149 p^{3} T^{4} + 872 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 306 T + 1078695 T^{2} - 213425084 T^{3} + 1078695 p^{3} T^{4} - 306 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1814 T + 2300598 T^{2} + 1795138900 T^{3} + 2300598 p^{3} T^{4} + 1814 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1764 T + 2238417 T^{2} - 1948100184 T^{3} + 2238417 p^{3} T^{4} - 1764 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 78 T + 934707 T^{2} + 79505188 T^{3} + 934707 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1216 T + 1698524 T^{2} + 1072507114 T^{3} + 1698524 p^{3} T^{4} + 1216 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.605436028713060230447721538337, −8.159072873967596067260241144677, −7.996005261524737656996004604284, −7.62094646975495302003257187109, −7.43937455300065812098903200880, −7.37424084885244425150116321756, −6.92228369087528057903018108288, −6.55791167464095470137693194248, −6.43147723816569084815235657581, −5.92386731170755028842886106282, −5.67291049076983454461262004472, −5.31153770247522109097384643710, −5.26937276707058206451927194757, −4.73391060355591954487196549793, −4.72753205038333068334900392431, −4.41202207398533501509739112362, −3.93873857447658962789739932634, −3.49861432933514688529916072892, −3.37770487741879905127637349741, −2.56030623003710973268822947649, −2.53566745888136690727770994463, −2.28922348216854036278405498713, −1.76842854810652235613633180482, −1.34157961487576080775559959375, −1.03003577304131052854734584336, 0, 0, 0,
1.03003577304131052854734584336, 1.34157961487576080775559959375, 1.76842854810652235613633180482, 2.28922348216854036278405498713, 2.53566745888136690727770994463, 2.56030623003710973268822947649, 3.37770487741879905127637349741, 3.49861432933514688529916072892, 3.93873857447658962789739932634, 4.41202207398533501509739112362, 4.72753205038333068334900392431, 4.73391060355591954487196549793, 5.26937276707058206451927194757, 5.31153770247522109097384643710, 5.67291049076983454461262004472, 5.92386731170755028842886106282, 6.43147723816569084815235657581, 6.55791167464095470137693194248, 6.92228369087528057903018108288, 7.37424084885244425150116321756, 7.43937455300065812098903200880, 7.62094646975495302003257187109, 7.996005261524737656996004604284, 8.159072873967596067260241144677, 8.605436028713060230447721538337
