| L(s) = 1 | − 2-s − 3·4-s + 3·5-s − 12·7-s + 4·8-s − 3·10-s + 6·11-s + 12·14-s + 3·16-s − 15·17-s + 19-s − 9·20-s − 6·22-s + 3·23-s + 6·25-s + 36·28-s − 2·29-s + 13·31-s − 6·32-s + 15·34-s − 36·35-s + 2·37-s − 38-s + 12·40-s − 6·41-s + 2·43-s − 18·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3/2·4-s + 1.34·5-s − 4.53·7-s + 1.41·8-s − 0.948·10-s + 1.80·11-s + 3.20·14-s + 3/4·16-s − 3.63·17-s + 0.229·19-s − 2.01·20-s − 1.27·22-s + 0.625·23-s + 6/5·25-s + 6.80·28-s − 0.371·29-s + 2.33·31-s − 1.06·32-s + 2.57·34-s − 6.08·35-s + 0.328·37-s − 0.162·38-s + 1.89·40-s − 0.937·41-s + 0.304·43-s − 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 17 T^{2} - 28 T^{3} + 17 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 15 T + 7 p T^{2} + 593 T^{3} + 7 p^{2} T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - T + 13 T^{2} + 89 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + p T^{2} - 41 T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_6$ | \( 1 + 2 T + 51 T^{2} + 108 T^{3} + 51 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 13 T + 105 T^{2} - 583 T^{3} + 105 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 2 T - 9 T^{2} + 196 T^{3} - 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 276 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 2 T + 65 T^{2} + 60 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 105 T^{2} - 873 T^{3} + 105 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - T + 101 T^{2} - 119 T^{3} + 101 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 1596 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 9 T + 161 T^{2} - 1069 T^{3} + 161 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 53 T^{2} - 160 T^{3} + 53 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 4 T + 181 T^{2} + 504 T^{3} + 181 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 4 T + 215 T^{2} + 576 T^{3} + 215 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 19 T + 285 T^{2} + 2653 T^{3} + 285 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 11 T + 203 T^{2} + 1867 T^{3} + 203 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 12 T + 231 T^{2} + 1920 T^{3} + 231 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 20 T + 415 T^{2} - 4112 T^{3} + 415 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.03919485549695784241187376218, −6.92781385078071678937731539372, −6.74463061136931774269186503854, −6.63963476205521154352416039535, −6.31776468449709642497610278420, −6.26112252148751132656557465889, −6.25407657055017165945267753272, −5.72316708073071230833919436196, −5.57897628140360803884546521714, −5.28722098685324247169890469343, −4.77765491573244976979147487098, −4.65602603618543809541210044412, −4.43642775208533803047423978477, −4.05211855016842051337082823228, −3.91071103250602498747809258022, −3.73889384833704419925650697321, −3.41386419043738701643663282976, −3.21902491428633574642340557123, −2.60076113971226188369106729342, −2.58816694958621094452525308425, −2.37994831550453648784706006307, −2.19778253531805911406355116416, −1.29359667207414693123325955088, −1.05949942705748576116678424625, −0.884380182825092990153487590383, 0, 0, 0,
0.884380182825092990153487590383, 1.05949942705748576116678424625, 1.29359667207414693123325955088, 2.19778253531805911406355116416, 2.37994831550453648784706006307, 2.58816694958621094452525308425, 2.60076113971226188369106729342, 3.21902491428633574642340557123, 3.41386419043738701643663282976, 3.73889384833704419925650697321, 3.91071103250602498747809258022, 4.05211855016842051337082823228, 4.43642775208533803047423978477, 4.65602603618543809541210044412, 4.77765491573244976979147487098, 5.28722098685324247169890469343, 5.57897628140360803884546521714, 5.72316708073071230833919436196, 6.25407657055017165945267753272, 6.26112252148751132656557465889, 6.31776468449709642497610278420, 6.63963476205521154352416039535, 6.74463061136931774269186503854, 6.92781385078071678937731539372, 7.03919485549695784241187376218
