| L(s) = 1 | − 2·2-s − 3·3-s − 4-s + 6·6-s + 3·7-s + 5·8-s + 6·9-s − 11-s + 3·12-s − 6·14-s − 16-s + 2·17-s − 12·18-s + 6·19-s − 9·21-s + 2·22-s + 23-s − 15·24-s − 15·25-s − 10·27-s − 3·28-s − 9·29-s + 10·31-s − 4·32-s + 3·33-s − 4·34-s − 6·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 1/2·4-s + 2.44·6-s + 1.13·7-s + 1.76·8-s + 2·9-s − 0.301·11-s + 0.866·12-s − 1.60·14-s − 1/4·16-s + 0.485·17-s − 2.82·18-s + 1.37·19-s − 1.96·21-s + 0.426·22-s + 0.208·23-s − 3.06·24-s − 3·25-s − 1.92·27-s − 0.566·28-s − 1.67·29-s + 1.79·31-s − 0.707·32-s + 0.522·33-s − 0.685·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 31 T^{2} + 21 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 2 T + 43 T^{2} - 60 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 124 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - T + 11 T^{2} - 59 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 9 T + 93 T^{2} + 479 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 10 T + 117 T^{2} - 628 T^{3} + 117 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 7 T + 97 T^{2} + 427 T^{3} + 97 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 20 T + 247 T^{2} + 1872 T^{3} + 247 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 13 T + 169 T^{2} - 1147 T^{3} + 169 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 57 T^{2} - 56 T^{3} + 57 p T^{4} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 5 T + 67 T^{2} + 153 T^{3} + 67 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 4 T + 145 T^{2} + 408 T^{3} + 145 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 4 T + 67 T^{2} - 592 T^{3} + 67 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 19 T + 305 T^{2} - 2673 T^{3} + 305 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 227 T^{2} + 1001 T^{3} + 227 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 14 T + 163 T^{2} + 1316 T^{3} + 163 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 173 T^{2} + 1379 T^{3} + 173 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 2 T + 185 T^{2} + 100 T^{3} + 185 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 183 T^{2} + 56 T^{3} + 183 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 8 T + 247 T^{2} - 1208 T^{3} + 247 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.949786454235079615998526346311, −7.82464450422104264392474080330, −7.70165970117943151204416329535, −7.34709074587352586217005365215, −6.93318401538512404147317053965, −6.72202446556602430024158907638, −6.63302926677696519108382941903, −6.18079759004290643897250482725, −5.77854997653464892852428672654, −5.55243859185269891026419905159, −5.34883723814263430799448936418, −5.26693072721926270555093662862, −5.24536222734334049546800961275, −4.51298159164695772891251737250, −4.40292752971199819964134375934, −4.27527623387980195449518462067, −3.81625613865847604911756365546, −3.62907568095711205136553374572, −3.28756720488356355281657739524, −2.74122590495942078563236848712, −2.30184813324046765327205025885, −1.95876004728810180744956589043, −1.31821916125168187748614679411, −1.22626367335038316231087646450, −1.18233350143246503856732644428, 0, 0, 0,
1.18233350143246503856732644428, 1.22626367335038316231087646450, 1.31821916125168187748614679411, 1.95876004728810180744956589043, 2.30184813324046765327205025885, 2.74122590495942078563236848712, 3.28756720488356355281657739524, 3.62907568095711205136553374572, 3.81625613865847604911756365546, 4.27527623387980195449518462067, 4.40292752971199819964134375934, 4.51298159164695772891251737250, 5.24536222734334049546800961275, 5.26693072721926270555093662862, 5.34883723814263430799448936418, 5.55243859185269891026419905159, 5.77854997653464892852428672654, 6.18079759004290643897250482725, 6.63302926677696519108382941903, 6.72202446556602430024158907638, 6.93318401538512404147317053965, 7.34709074587352586217005365215, 7.70165970117943151204416329535, 7.82464450422104264392474080330, 7.949786454235079615998526346311
