| L(s) = 1 | − 2-s + 3·3-s − 3·6-s − 7-s + 6·9-s − 11-s + 14-s − 6·18-s − 19-s − 3·21-s + 22-s − 23-s + 3·25-s + 10·27-s − 31-s − 3·33-s − 37-s + 38-s − 41-s + 3·42-s + 46-s − 3·50-s − 10·54-s − 3·57-s − 59-s − 61-s + 62-s + ⋯ |
| L(s) = 1 | − 2-s + 3·3-s − 3·6-s − 7-s + 6·9-s − 11-s + 14-s − 6·18-s − 19-s − 3·21-s + 22-s − 23-s + 3·25-s + 10·27-s − 31-s − 3·33-s − 37-s + 38-s − 41-s + 3·42-s + 46-s − 3·50-s − 10·54-s − 3·57-s − 59-s − 61-s + 62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89314623 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89314623 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7731812405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7731812405\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 149 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 67 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 71 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 73 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−10.14866620493946758315578061031, −9.614244169881286737611016439084, −9.307110975782005013404183436806, −9.250220096751762148816611936882, −8.724622655345588025712028427955, −8.714850795603752631392194394422, −8.590467746275215297993492331941, −8.077015532215392242406297269297, −7.906018658857740387382729385851, −7.44746529242760733404610266845, −7.24297226166409964931876886576, −6.89583229976468280121918581623, −6.50757133247180643878474415880, −6.43635493766420963173751838123, −5.58398827218719397284982852382, −5.18981868377537442811086082520, −4.68282744444178734220977853276, −4.22631123461919212976758328393, −4.16629376621955412865319929053, −3.35872673140665124446696475988, −3.21094576969232543260813648216, −2.78319217042432333593650012333, −2.64789455287754599161843283869, −1.76457432776260659837964002510, −1.56232957238509874094185475666,
1.56232957238509874094185475666, 1.76457432776260659837964002510, 2.64789455287754599161843283869, 2.78319217042432333593650012333, 3.21094576969232543260813648216, 3.35872673140665124446696475988, 4.16629376621955412865319929053, 4.22631123461919212976758328393, 4.68282744444178734220977853276, 5.18981868377537442811086082520, 5.58398827218719397284982852382, 6.43635493766420963173751838123, 6.50757133247180643878474415880, 6.89583229976468280121918581623, 7.24297226166409964931876886576, 7.44746529242760733404610266845, 7.906018658857740387382729385851, 8.077015532215392242406297269297, 8.590467746275215297993492331941, 8.714850795603752631392194394422, 8.724622655345588025712028427955, 9.250220096751762148816611936882, 9.307110975782005013404183436806, 9.614244169881286737611016439084, 10.14866620493946758315578061031
