| L(s) = 1 | + 2-s − 3·3-s − 3·6-s + 2·8-s + 6·9-s + 3·11-s + 2·13-s + 3·16-s + 2·17-s + 6·18-s + 8·19-s + 3·22-s − 6·24-s + 2·26-s − 10·27-s − 10·29-s + 8·31-s + 3·32-s − 9·33-s + 2·34-s + 6·37-s + 8·38-s − 6·39-s − 14·41-s − 4·43-s + 8·47-s − 9·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1.22·6-s + 0.707·8-s + 2·9-s + 0.904·11-s + 0.554·13-s + 3/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s + 0.639·22-s − 1.22·24-s + 0.392·26-s − 1.92·27-s − 1.85·29-s + 1.43·31-s + 0.530·32-s − 1.56·33-s + 0.342·34-s + 0.986·37-s + 1.29·38-s − 0.960·39-s − 2.18·41-s − 0.609·43-s + 1.16·47-s − 1.29·48-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
| \(L(1)\) |
\(\approx\) |
\(2.743864766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743864766\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T - T^{2} + 116 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 14 T + 167 T^{2} + 1156 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 56 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 107 T^{2} - 644 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1096 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 153 T^{2} - 472 T^{3} + 153 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 181 T^{2} - 1008 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 223 T^{2} - 1700 T^{3} + 223 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 173 T^{2} - 1096 T^{3} + 173 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} - 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 215 T^{2} + 1580 T^{3} + 215 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 399 T^{2} + 4276 T^{3} + 399 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.509729547902967316931157586557, −8.719332441510926997684578857334, −8.409578853511103745454259503108, −8.318272012970896134030915355471, −7.80148911629081451534797706316, −7.58644271444495996613638756312, −7.26252456607141353211515782971, −6.86355015065206450571995951020, −6.65996652849161641384712338128, −6.61067391875904416570933614709, −5.82175593387592209920033989527, −5.79551129756285340068851740203, −5.66318824732485087487866037554, −5.04109463830870922650412143532, −4.88109582633404357410849875738, −4.87616480452632839857544481654, −3.93947098832744351545324993545, −3.93011457200404357013828179400, −3.90153366285414162564750922574, −3.10743837981281914408068596125, −2.86908830676011797181091771367, −1.91027492340035911089610934674, −1.67763356661225371788350724459, −1.01429119232244115145294490384, −0.72488056187630882142132887690,
0.72488056187630882142132887690, 1.01429119232244115145294490384, 1.67763356661225371788350724459, 1.91027492340035911089610934674, 2.86908830676011797181091771367, 3.10743837981281914408068596125, 3.90153366285414162564750922574, 3.93011457200404357013828179400, 3.93947098832744351545324993545, 4.87616480452632839857544481654, 4.88109582633404357410849875738, 5.04109463830870922650412143532, 5.66318824732485087487866037554, 5.79551129756285340068851740203, 5.82175593387592209920033989527, 6.61067391875904416570933614709, 6.65996652849161641384712338128, 6.86355015065206450571995951020, 7.26252456607141353211515782971, 7.58644271444495996613638756312, 7.80148911629081451534797706316, 8.318272012970896134030915355471, 8.409578853511103745454259503108, 8.719332441510926997684578857334, 9.509729547902967316931157586557
