| L(s) = 1 | − 6·2-s + 3·3-s + 24·4-s − 8·5-s − 18·6-s − 80·8-s − 42·9-s + 48·10-s + 6·11-s + 72·12-s − 13·13-s − 24·15-s + 240·16-s − 20·17-s + 252·18-s + 12·19-s − 192·20-s − 36·22-s − 69·23-s − 240·24-s − 177·25-s + 78·26-s − 163·27-s − 67·29-s + 144·30-s − 41·31-s − 672·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3·4-s − 0.715·5-s − 1.22·6-s − 3.53·8-s − 1.55·9-s + 1.51·10-s + 0.164·11-s + 1.73·12-s − 0.277·13-s − 0.413·15-s + 15/4·16-s − 0.285·17-s + 3.29·18-s + 0.144·19-s − 2.14·20-s − 0.348·22-s − 0.625·23-s − 2.04·24-s − 1.41·25-s + 0.588·26-s − 1.16·27-s − 0.429·29-s + 0.876·30-s − 0.237·31-s − 3.71·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - p T + 17 p T^{2} - 116 T^{3} + 17 p^{4} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 8 T + 241 T^{2} + 2336 T^{3} + 241 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 3477 T^{2} - 15820 T^{3} + 3477 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + p T + 5743 T^{2} + 48890 T^{3} + 5743 p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 20 T + 14337 T^{2} + 197288 T^{3} + 14337 p^{3} T^{4} + 20 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 12 T + 7653 T^{2} - 524904 T^{3} + 7653 p^{3} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 67 T + 6079 T^{2} - 2791802 T^{3} + 6079 p^{3} T^{4} + 67 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 41 T + 17263 T^{2} - 99032 T^{3} + 17263 p^{3} T^{4} + 41 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 36 T + 109539 T^{2} + 5834088 T^{3} + 109539 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 273 T + 220107 T^{2} + 36881094 T^{3} + 220107 p^{3} T^{4} + 273 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 564 T + 327609 T^{2} - 92381880 T^{3} + 327609 p^{3} T^{4} - 564 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 1145 T + 697887 T^{2} + 272530952 T^{3} + 697887 p^{3} T^{4} + 1145 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 632 T + 507843 T^{2} - 183016256 T^{3} + 507843 p^{3} T^{4} - 632 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 128 T + 201511 T^{2} + 67524280 T^{3} + 201511 p^{3} T^{4} - 128 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 266 T + 630617 T^{2} + 110916680 T^{3} + 630617 p^{3} T^{4} + 266 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1128 T + 1285221 T^{2} - 715046656 T^{3} + 1285221 p^{3} T^{4} - 1128 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1075 T + 1422925 T^{2} - 800571106 T^{3} + 1422925 p^{3} T^{4} - 1075 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 59 T + 162635 T^{2} + 251207930 T^{3} + 162635 p^{3} T^{4} + 59 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 744 T + 869673 T^{2} + 635830144 T^{3} + 869673 p^{3} T^{4} + 744 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2708 T + 3903245 T^{2} + 3586323960 T^{3} + 3903245 p^{3} T^{4} + 2708 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 448 T + 1612617 T^{2} + 707661976 T^{3} + 1612617 p^{3} T^{4} + 448 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1322 T + 1040633 T^{2} - 1001644904 T^{3} + 1040633 p^{3} T^{4} - 1322 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.233415677265477648799570996711, −7.88427259289301532109579215507, −7.55092493200683228694329383046, −7.45142130696493152047355657719, −7.35302866930895167404280226921, −6.78985885669453201253627076053, −6.65392230687402623552823791410, −6.19623599985866653147637043863, −6.16010869251930231255385033805, −5.89063144699798236821054311618, −5.33768757729103878498111881608, −5.25126733432022494595889936339, −5.07360452562253318253967715184, −4.26827109747877478883847438228, −4.18958020458546189922085567358, −3.76541378859924648419105780743, −3.40942586222233113961800734360, −3.24651973965303865542739117333, −2.93401365491670666738430605025, −2.37347507700532321963930284678, −2.19069174526258483468728586420, −2.13786317927966686125655223012, −1.49039631562347948871697109185, −1.18183834997668096735039076969, −0.70250433787796759767131451223, 0, 0, 0,
0.70250433787796759767131451223, 1.18183834997668096735039076969, 1.49039631562347948871697109185, 2.13786317927966686125655223012, 2.19069174526258483468728586420, 2.37347507700532321963930284678, 2.93401365491670666738430605025, 3.24651973965303865542739117333, 3.40942586222233113961800734360, 3.76541378859924648419105780743, 4.18958020458546189922085567358, 4.26827109747877478883847438228, 5.07360452562253318253967715184, 5.25126733432022494595889936339, 5.33768757729103878498111881608, 5.89063144699798236821054311618, 6.16010869251930231255385033805, 6.19623599985866653147637043863, 6.65392230687402623552823791410, 6.78985885669453201253627076053, 7.35302866930895167404280226921, 7.45142130696493152047355657719, 7.55092493200683228694329383046, 7.88427259289301532109579215507, 8.233415677265477648799570996711
