| L(s) = 1 | − 2-s + 3·3-s − 3·6-s + 4·7-s + 8-s + 6·9-s + 9·11-s − 6·13-s − 4·14-s − 3·16-s − 2·17-s − 6·18-s − 4·19-s + 12·21-s − 9·22-s − 23-s + 3·24-s + 6·26-s + 10·27-s − 3·29-s + 27·33-s + 2·34-s − 13·37-s + 4·38-s − 18·39-s + 13·41-s − 12·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1.22·6-s + 1.51·7-s + 0.353·8-s + 2·9-s + 2.71·11-s − 1.66·13-s − 1.06·14-s − 3/4·16-s − 0.485·17-s − 1.41·18-s − 0.917·19-s + 2.61·21-s − 1.91·22-s − 0.208·23-s + 0.612·24-s + 1.17·26-s + 1.92·27-s − 0.557·29-s + 4.70·33-s + 0.342·34-s − 2.13·37-s + 0.648·38-s − 2.88·39-s + 2.03·41-s − 1.85·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.403389577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.403389577\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 2 p T^{2} - 6 p T^{3} + 2 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 38 T^{2} + 154 T^{3} + 38 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 40 T^{2} + 60 T^{3} + 40 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 64 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} - 66 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 706 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 13 T + 167 T^{2} - 1094 T^{3} + 167 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 13 T + 131 T^{2} - 810 T^{3} + 131 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 72 T^{2} - 78 T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 45 T^{2} - 634 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 22 T + 317 T^{2} - 2852 T^{3} + 317 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 187 T^{2} - 1108 T^{3} + 187 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 28 T + 406 T^{2} - 3946 T^{3} + 406 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 488 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 35 T^{2} - 10 p T^{3} + 35 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 97 T^{2} + 92 T^{3} + 97 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 311 T^{2} - 2534 T^{3} + 311 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 450 T^{2} - 4738 T^{3} + 450 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - T + 149 T^{2} - 270 T^{3} + 149 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.246182014506372122085848519962, −7.67669623247393556519258627825, −7.63948407361398088025771493507, −7.43359338397312066616692158129, −7.09860990898367678667105539683, −6.92096912349969274890568152242, −6.56543446671410866068438699800, −6.44737959372255806264854839653, −6.04789831851620209930441203112, −5.63238044193880760565239356134, −5.04168403094639554591215467887, −4.95121189473724260167217003058, −4.92866357742522007968505500261, −4.16522623648030700895794611962, −4.12035256243533783564754959823, −3.99969796799729332953802914103, −3.61887807605568899539985127213, −3.39693749655534156302946259097, −2.66993356258791298783687593554, −2.30807327490443991618095563779, −2.06537309341979483712572173237, −2.01969015414635785340615067494, −1.60053996753598117862128092774, −0.815274929991884740931393064079, −0.78318140584315882525102430039,
0.78318140584315882525102430039, 0.815274929991884740931393064079, 1.60053996753598117862128092774, 2.01969015414635785340615067494, 2.06537309341979483712572173237, 2.30807327490443991618095563779, 2.66993356258791298783687593554, 3.39693749655534156302946259097, 3.61887807605568899539985127213, 3.99969796799729332953802914103, 4.12035256243533783564754959823, 4.16522623648030700895794611962, 4.92866357742522007968505500261, 4.95121189473724260167217003058, 5.04168403094639554591215467887, 5.63238044193880760565239356134, 6.04789831851620209930441203112, 6.44737959372255806264854839653, 6.56543446671410866068438699800, 6.92096912349969274890568152242, 7.09860990898367678667105539683, 7.43359338397312066616692158129, 7.63948407361398088025771493507, 7.67669623247393556519258627825, 8.246182014506372122085848519962
