| L(s) = 1 | − 3·3-s + 8-s + 6·9-s − 3·11-s − 12·13-s + 6·17-s − 3·19-s + 9·23-s − 3·24-s − 10·27-s − 15·29-s + 15·31-s + 9·33-s − 12·37-s + 36·39-s + 12·41-s − 12·43-s − 12·47-s − 18·51-s + 9·53-s + 9·57-s + 12·59-s + 3·61-s − 7·64-s − 15·67-s − 27·69-s + 18·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.353·8-s + 2·9-s − 0.904·11-s − 3.32·13-s + 1.45·17-s − 0.688·19-s + 1.87·23-s − 0.612·24-s − 1.92·27-s − 2.78·29-s + 2.69·31-s + 1.56·33-s − 1.97·37-s + 5.76·39-s + 1.87·41-s − 1.82·43-s − 1.75·47-s − 2.52·51-s + 1.23·53-s + 1.19·57-s + 1.56·59-s + 0.384·61-s − 7/8·64-s − 1.83·67-s − 3.25·69-s + 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7016594288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7016594288\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - T^{3} + p^{3} T^{6} \) |
| 7 | $D_{6}$ | \( 1 - 16 T^{3} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 50 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 15 T^{2} + 4 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 75 T^{2} - 362 T^{3} + 75 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 15 T + 153 T^{2} - 978 T^{3} + 153 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 135 T^{2} + 864 T^{3} + 135 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 156 T^{2} - 990 T^{3} + 156 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 129 T^{2} + 936 T^{3} + 129 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 90 T^{2} - 757 T^{3} + 90 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1096 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 126 T^{2} - 323 T^{3} + 126 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 261 T^{2} + 2062 T^{3} + 261 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 18 T + 264 T^{2} - 2274 T^{3} + 264 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 126 T^{2} + 407 T^{3} + 126 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} + 146 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3 T + 237 T^{2} + 486 T^{3} + 237 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 186 T^{2} + 579 T^{3} + 186 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 219 T^{2} - 1424 T^{3} + 219 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.387353669502539239257027717848, −7.955343051482554743636463591493, −7.909587430441578495246300647404, −7.57215368575894029225126272950, −7.30228493713421563642290511461, −7.00541146682420557332090771025, −7.00340272109765059116676305406, −6.59989513725105692016051600026, −6.32060936102094730119476188868, −5.69310278425617168497403709206, −5.67152466362370697840996444150, −5.33101823238736231735751297204, −5.15532639751351822631500293161, −4.75493741023994062665024257310, −4.66870305760039868359795825944, −4.54160591641743018432560801856, −3.83758430565656997567348317645, −3.48385403646453888532603394780, −3.15801417653304313122109978874, −2.54311255010881289997729599166, −2.49511161767803834173530802904, −1.84102821426157167039284467707, −1.54543612997203441825260157120, −0.70162405211992831969136414208, −0.36753422753951969806669635017,
0.36753422753951969806669635017, 0.70162405211992831969136414208, 1.54543612997203441825260157120, 1.84102821426157167039284467707, 2.49511161767803834173530802904, 2.54311255010881289997729599166, 3.15801417653304313122109978874, 3.48385403646453888532603394780, 3.83758430565656997567348317645, 4.54160591641743018432560801856, 4.66870305760039868359795825944, 4.75493741023994062665024257310, 5.15532639751351822631500293161, 5.33101823238736231735751297204, 5.67152466362370697840996444150, 5.69310278425617168497403709206, 6.32060936102094730119476188868, 6.59989513725105692016051600026, 7.00340272109765059116676305406, 7.00541146682420557332090771025, 7.30228493713421563642290511461, 7.57215368575894029225126272950, 7.909587430441578495246300647404, 7.955343051482554743636463591493, 8.387353669502539239257027717848
