| L(s) = 1 | − 2-s − 3·3-s − 2·4-s + 3·6-s − 4·7-s + 2·8-s + 6·9-s + 3·11-s + 6·12-s − 2·13-s + 4·14-s + 16-s − 6·18-s − 6·19-s + 12·21-s − 3·22-s − 12·23-s − 6·24-s + 2·26-s − 10·27-s + 8·28-s + 8·29-s − 8·31-s + 32-s − 9·33-s − 12·36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 1.51·7-s + 0.707·8-s + 2·9-s + 0.904·11-s + 1.73·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.41·18-s − 1.37·19-s + 2.61·21-s − 0.639·22-s − 2.50·23-s − 1.22·24-s + 0.392·26-s − 1.92·27-s + 1.51·28-s + 1.48·29-s − 1.43·31-s + 0.176·32-s − 1.56·33-s − 2·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 52 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 48 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 23 T^{2} - 52 T^{3} + 23 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 - 8 T + 71 T^{2} - 304 T^{3} + 71 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 101 T^{2} + 480 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 95 T^{2} + 264 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 11 T^{2} + 272 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 884 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 189 T^{2} + 1536 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 191 T^{2} + 1712 T^{3} + 191 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 113 T^{2} - 1024 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 291 T^{2} + 2692 T^{3} + 291 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 185 T^{2} + 1544 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 117 T^{2} + 760 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 175 T^{2} - 1072 T^{3} + 175 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 10 T + 25 T^{2} - 140 T^{3} + 25 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 189 T^{2} - 1056 T^{3} + 189 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 255 T^{2} - 2684 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 16 T + 131 T^{2} + 672 T^{3} + 131 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.667750402935049765515411567581, −9.255685963136796812018701128104, −9.083657907431766356766631054875, −8.958755080134150296825081630491, −8.214905430638894958424429450593, −8.152398093026085170495732770405, −8.068013884955445269719119145717, −7.39033607072128127597643024710, −7.27222957457784295880375963242, −6.66540924706850267965456337882, −6.41431558191273704550629764140, −6.28561739412942160519080176542, −6.28062773744403945913256246318, −5.88651321407098908231315876578, −5.24767170607294390761269817532, −5.09996759719112183530139638036, −4.64017506545693089220384394903, −4.38568330778879800444946357376, −4.24391503015009162106945243657, −3.61582496401322644909572348923, −3.41778385349178823269438616811, −2.97910414254756085509959498399, −2.22506912740385525617645534196, −1.63690961981755026140006473248, −1.36951501118679666259017721234, 0, 0, 0,
1.36951501118679666259017721234, 1.63690961981755026140006473248, 2.22506912740385525617645534196, 2.97910414254756085509959498399, 3.41778385349178823269438616811, 3.61582496401322644909572348923, 4.24391503015009162106945243657, 4.38568330778879800444946357376, 4.64017506545693089220384394903, 5.09996759719112183530139638036, 5.24767170607294390761269817532, 5.88651321407098908231315876578, 6.28062773744403945913256246318, 6.28561739412942160519080176542, 6.41431558191273704550629764140, 6.66540924706850267965456337882, 7.27222957457784295880375963242, 7.39033607072128127597643024710, 8.068013884955445269719119145717, 8.152398093026085170495732770405, 8.214905430638894958424429450593, 8.958755080134150296825081630491, 9.083657907431766356766631054875, 9.255685963136796812018701128104, 9.667750402935049765515411567581
