| L(s) = 1 | − 3·2-s + 6·4-s + 8·5-s + 3·7-s − 10·8-s − 2·9-s − 24·10-s + 9·11-s − 9·14-s + 15·16-s + 10·17-s + 6·18-s + 7·19-s + 48·20-s − 27·22-s + 3·23-s + 30·25-s − 7·27-s + 18·28-s + 5·29-s + 24·31-s − 21·32-s − 30·34-s + 24·35-s − 12·36-s − 4·37-s − 21·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 3.57·5-s + 1.13·7-s − 3.53·8-s − 2/3·9-s − 7.58·10-s + 2.71·11-s − 2.40·14-s + 15/4·16-s + 2.42·17-s + 1.41·18-s + 1.60·19-s + 10.7·20-s − 5.75·22-s + 0.625·23-s + 6·25-s − 1.34·27-s + 3.40·28-s + 0.928·29-s + 4.31·31-s − 3.71·32-s − 5.14·34-s + 4.05·35-s − 2·36-s − 0.657·37-s − 3.40·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.015378700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.015378700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 T^{2} + 7 T^{3} + 2 p T^{4} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 8 T + 34 T^{2} - 93 T^{3} + 34 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_6$ | \( 1 - 9 T + 53 T^{2} - 211 T^{3} + 53 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 10 T + 40 T^{2} - 117 T^{3} + 40 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 7 T + 3 p T^{2} - 259 T^{3} + 3 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 65 T^{2} - 125 T^{3} + 65 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 5 T - 5 T^{2} + 3 p T^{3} - 5 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 1951 T^{3} + 278 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 4 T + 72 T^{2} + 337 T^{3} + 72 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 77 T^{2} + 289 T^{3} + 77 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 9 T + 149 T^{2} + 787 T^{3} + 149 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 17 T + 5 p T^{2} - 1767 T^{3} + 5 p^{2} T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 10 T + 176 T^{2} + 1019 T^{3} + 176 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 3 T + 131 T^{2} + 355 T^{3} + 131 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 5 T + 105 T^{2} + 779 T^{3} + 105 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 5 T + 95 T^{2} + 867 T^{3} + 95 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 24 T + 356 T^{2} - 3577 T^{3} + 356 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - T + 189 T^{2} - 103 T^{3} + 189 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 10 T + 128 T^{2} - 489 T^{3} + 128 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 95 T^{2} + 371 T^{3} + 95 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 2 T + 252 T^{2} - 327 T^{3} + 252 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 27 T + 527 T^{2} + 5911 T^{3} + 527 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.238471729055428284484032593621, −7.74406930160907968527565867532, −7.58335892993246119175153945267, −7.19597749223697495551133631740, −6.90842359133185767686178381146, −6.59693818746280983476846646540, −6.39966943632074362453496766591, −6.18237686615607193390834959819, −5.98419674249783702688307954685, −5.86864710004429638845260910803, −5.32477366607071113467159620445, −5.26269504056444984751283714102, −5.05438747055774578017143295778, −4.61711168714485087558550050286, −4.01093923181548279021059987007, −3.70212808866987808248794180483, −3.22767522512921899371598752269, −2.89174820202305753900246619076, −2.69625693791264618112334782375, −2.28148298856700001697264545153, −1.91381429650179824514052015099, −1.54457605411617663286612925115, −1.16080912754010559199178432945, −1.12210339714775133250868840579, −1.02202240437057255321323684035,
1.02202240437057255321323684035, 1.12210339714775133250868840579, 1.16080912754010559199178432945, 1.54457605411617663286612925115, 1.91381429650179824514052015099, 2.28148298856700001697264545153, 2.69625693791264618112334782375, 2.89174820202305753900246619076, 3.22767522512921899371598752269, 3.70212808866987808248794180483, 4.01093923181548279021059987007, 4.61711168714485087558550050286, 5.05438747055774578017143295778, 5.26269504056444984751283714102, 5.32477366607071113467159620445, 5.86864710004429638845260910803, 5.98419674249783702688307954685, 6.18237686615607193390834959819, 6.39966943632074362453496766591, 6.59693818746280983476846646540, 6.90842359133185767686178381146, 7.19597749223697495551133631740, 7.58335892993246119175153945267, 7.74406930160907968527565867532, 8.238471729055428284484032593621
