| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s + 9·6-s + 8·7-s + 4·8-s + 6·9-s + 12·12-s + 6·13-s + 24·14-s + 3·16-s + 4·17-s + 18·18-s + 2·19-s + 24·21-s − 12·23-s + 12·24-s + 18·26-s + 10·27-s + 32·28-s + 8·29-s + 8·31-s − 32-s + 12·34-s + 24·36-s − 4·37-s + 6·38-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s + 3.67·6-s + 3.02·7-s + 1.41·8-s + 2·9-s + 3.46·12-s + 1.66·13-s + 6.41·14-s + 3/4·16-s + 0.970·17-s + 4.24·18-s + 0.458·19-s + 5.23·21-s − 2.50·23-s + 2.44·24-s + 3.53·26-s + 1.92·27-s + 6.04·28-s + 1.48·29-s + 1.43·31-s − 0.176·32-s + 2.05·34-s + 4·36-s − 0.657·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(104.0981210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(104.0981210\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 20 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} + 108 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 432 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 624 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 145 T^{2} - 692 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 125 T^{2} + 592 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 127 T^{2} - 576 T^{3} + 127 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 2 T + 131 T^{2} + 204 T^{3} + 131 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 185 T^{2} + 1288 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 279 T^{2} - 2536 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 233 T^{2} - 940 T^{3} + 233 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 245 T^{2} + 328 T^{3} + 245 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 255 T^{2} - 348 T^{3} + 255 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 259 T^{2} - 1424 T^{3} + 259 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.72176784957749855626729729711, −6.61325391178579816190892295001, −6.15729711952447827858920300835, −6.04263812970950156136323919279, −5.82211335118011540854835073926, −5.51720600073119788253420642123, −5.50010325320873716703442156154, −4.88869924655958038353473046937, −4.79002671290631610833740203557, −4.63578581259931773563150557616, −4.41398244738947921079902727578, −4.29159704524203784988612238903, −4.22852520169071384707905396936, −3.58088400067104101410860796764, −3.51000449289901905732841120458, −3.48285827296203539637603309202, −2.96850132261302279008297676182, −2.86039017852520544331053773583, −2.21355586031351909025284776293, −2.12674544853775588197824821184, −1.97671941444564389074796256942, −1.64638984761595203001268343530, −1.32750749544657828394095531713, −0.873204115582791448545432409592, −0.833837575694138690640980516308,
0.833837575694138690640980516308, 0.873204115582791448545432409592, 1.32750749544657828394095531713, 1.64638984761595203001268343530, 1.97671941444564389074796256942, 2.12674544853775588197824821184, 2.21355586031351909025284776293, 2.86039017852520544331053773583, 2.96850132261302279008297676182, 3.48285827296203539637603309202, 3.51000449289901905732841120458, 3.58088400067104101410860796764, 4.22852520169071384707905396936, 4.29159704524203784988612238903, 4.41398244738947921079902727578, 4.63578581259931773563150557616, 4.79002671290631610833740203557, 4.88869924655958038353473046937, 5.50010325320873716703442156154, 5.51720600073119788253420642123, 5.82211335118011540854835073926, 6.04263812970950156136323919279, 6.15729711952447827858920300835, 6.61325391178579816190892295001, 6.72176784957749855626729729711
