| L(s) = 1 | − 3·3-s − 3·7-s + 6·9-s − 2·11-s + 12·13-s − 3·19-s + 9·21-s − 2·23-s − 5·25-s − 10·27-s + 2·29-s − 8·31-s + 6·33-s + 6·37-s − 36·39-s + 2·41-s − 4·43-s − 4·47-s + 6·49-s − 14·53-s + 9·57-s + 4·59-s + 2·61-s − 18·63-s + 2·67-s + 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 2·9-s − 0.603·11-s + 3.32·13-s − 0.688·19-s + 1.96·21-s − 0.417·23-s − 25-s − 1.92·27-s + 0.371·29-s − 1.43·31-s + 1.04·33-s + 0.986·37-s − 5.76·39-s + 0.312·41-s − 0.609·43-s − 0.583·47-s + 6/7·49-s − 1.92·53-s + 1.19·57-s + 0.520·59-s + 0.256·61-s − 2.26·63-s + 0.244·67-s + 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412201695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412201695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 + p T^{2} - 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 60 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 41 T^{2} - 8 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 45 T^{2} + 108 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} - 72 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 480 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 47 T^{2} - 364 T^{3} + 47 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 109 T^{2} + 280 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 127 T^{2} + 384 T^{3} + 127 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 173 T^{2} + 1480 T^{3} + 173 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 216 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 236 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 177 T^{2} - 284 T^{3} + 177 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 167 T^{2} - 1120 T^{3} + 167 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 207 T^{2} - 2036 T^{3} + 207 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 205 T^{2} - 284 T^{3} + 205 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 187 T^{2} - 432 T^{3} + 187 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 223 T^{2} - 1308 T^{3} + 223 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 519 T^{2} - 5944 T^{3} + 519 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.88317942641693012488783809610, −7.27375149529527425785430727529, −7.06296056686310212041801342550, −6.80432049964835768289099855367, −6.36302616047263812077496730309, −6.35792213678913352447124944374, −6.31199051359271210012986130471, −5.86627081615606313954444557162, −5.68539760818035997621903438228, −5.66517826480688032382770459141, −5.13389633341928851769926104181, −4.79143441719933114153996239098, −4.73925092924410617821929570878, −3.98857568308440947226259945649, −3.97893800930649243095073269994, −3.95587404589981668193774263159, −3.30757105671798526735996101808, −3.22855607241081731270632246762, −2.99960386275550820394931859100, −2.05514387879784275320835113347, −1.87957887275407220150839797997, −1.86302393491333755449469799399, −0.998220037424015437453987781646, −0.74904572332696076940314921049, −0.39441577959894301066409407001,
0.39441577959894301066409407001, 0.74904572332696076940314921049, 0.998220037424015437453987781646, 1.86302393491333755449469799399, 1.87957887275407220150839797997, 2.05514387879784275320835113347, 2.99960386275550820394931859100, 3.22855607241081731270632246762, 3.30757105671798526735996101808, 3.95587404589981668193774263159, 3.97893800930649243095073269994, 3.98857568308440947226259945649, 4.73925092924410617821929570878, 4.79143441719933114153996239098, 5.13389633341928851769926104181, 5.66517826480688032382770459141, 5.68539760818035997621903438228, 5.86627081615606313954444557162, 6.31199051359271210012986130471, 6.35792213678913352447124944374, 6.36302616047263812077496730309, 6.80432049964835768289099855367, 7.06296056686310212041801342550, 7.27375149529527425785430727529, 7.88317942641693012488783809610
