| L(s) = 1 | + 3·3-s − 3·4-s + 6·5-s − 3·7-s − 8-s + 6·9-s + 6·11-s − 9·12-s + 18·15-s + 3·16-s + 6·17-s + 9·19-s − 18·20-s − 9·21-s + 3·23-s − 3·24-s + 12·25-s + 10·27-s + 9·28-s + 9·29-s + 9·31-s + 6·32-s + 18·33-s − 18·35-s − 18·36-s + 12·37-s − 6·40-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 3/2·4-s + 2.68·5-s − 1.13·7-s − 0.353·8-s + 2·9-s + 1.80·11-s − 2.59·12-s + 4.64·15-s + 3/4·16-s + 1.45·17-s + 2.06·19-s − 4.02·20-s − 1.96·21-s + 0.625·23-s − 0.612·24-s + 12/5·25-s + 1.92·27-s + 1.70·28-s + 1.67·29-s + 1.61·31-s + 1.06·32-s + 3.13·33-s − 3.04·35-s − 3·36-s + 1.97·37-s − 0.948·40-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
| \(L(1)\) |
\(\approx\) |
\(17.40377548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.40377548\) |
| \(L(\frac{3}{2})\) |
|
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 | $A_4\times C_2$ | \( 1 + 3 T^{2} + T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 45 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 72 T^{2} - 325 T^{3} + 72 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 81 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 9 T + 39 T^{2} - 199 T^{3} + 39 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 531 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 12 T + 102 T^{2} - 561 T^{3} + 102 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 6 T - 9 T^{2} + 364 T^{3} - 9 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 81 T^{2} - 701 T^{3} + 81 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 279 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 549 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 15 T + 105 T^{2} - 477 T^{3} + 105 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 222 T^{2} + 1703 T^{3} + 222 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 9 T + 27 T^{2} + 377 T^{3} + 27 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 12 T + 222 T^{2} + 1593 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 30 T + 498 T^{2} - 5187 T^{3} + 498 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 21 T + 375 T^{2} + 3589 T^{3} + 375 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 24 T + 402 T^{2} - 4095 T^{3} + 402 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 9 T + 282 T^{2} + 1601 T^{3} + 282 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 3 T + 165 T^{2} - 525 T^{3} + 165 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84569392256303307706650885703, −7.39308716081996662779549351057, −7.01589626720278875304683750325, −6.80131947602697346797153626598, −6.52276669095203499306974763838, −6.21767162650072858822486909705, −6.20227158537072456289838948201, −5.81852869348851639860905073765, −5.58884854850225640300009425291, −5.38888808477002738332814005301, −4.95118638011673868747817695017, −4.78029320371243054188958918951, −4.42644780265192858675443408958, −4.01975977764998503121454585398, −3.89903890151244480712108289233, −3.65640963159534238552336320162, −3.22507822494089477627542525867, −2.86073994069308439431166431864, −2.78314041575731476360502797202, −2.39873661143261404725447621796, −2.27858558381304236900852184715, −1.59056020682012240095558292916, −1.05552969012468383006424373144, −1.00794629384530713874877279802, −0.905595143552427927379995013257,
0.905595143552427927379995013257, 1.00794629384530713874877279802, 1.05552969012468383006424373144, 1.59056020682012240095558292916, 2.27858558381304236900852184715, 2.39873661143261404725447621796, 2.78314041575731476360502797202, 2.86073994069308439431166431864, 3.22507822494089477627542525867, 3.65640963159534238552336320162, 3.89903890151244480712108289233, 4.01975977764998503121454585398, 4.42644780265192858675443408958, 4.78029320371243054188958918951, 4.95118638011673868747817695017, 5.38888808477002738332814005301, 5.58884854850225640300009425291, 5.81852869348851639860905073765, 6.20227158537072456289838948201, 6.21767162650072858822486909705, 6.52276669095203499306974763838, 6.80131947602697346797153626598, 7.01589626720278875304683750325, 7.39308716081996662779549351057, 7.84569392256303307706650885703
