| L(s) = 1 | − 3·2-s + 6·4-s − 5-s + 9·7-s − 10·8-s + 3·10-s − 5·11-s − 27·14-s + 15·16-s + 8·17-s − 4·19-s − 6·20-s + 15·22-s + 2·25-s + 54·28-s − 11·29-s + 5·31-s − 21·32-s − 24·34-s − 9·35-s − 8·37-s + 12·38-s + 10·40-s − 2·41-s + 12·43-s − 30·44-s − 4·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 0.447·5-s + 3.40·7-s − 3.53·8-s + 0.948·10-s − 1.50·11-s − 7.21·14-s + 15/4·16-s + 1.94·17-s − 0.917·19-s − 1.34·20-s + 3.19·22-s + 2/5·25-s + 10.2·28-s − 2.04·29-s + 0.898·31-s − 3.71·32-s − 4.11·34-s − 1.52·35-s − 1.31·37-s + 1.94·38-s + 1.58·40-s − 0.312·41-s + 1.82·43-s − 4.52·44-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.738370419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738370419\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + T - T^{2} - 19 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 9 T + 41 T^{2} - 125 T^{3} + 41 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 5 T + 25 T^{2} + 111 T^{3} + 25 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 8 T + 63 T^{2} - 264 T^{3} + 63 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 88 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 41 T^{2} - 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 11 T + 111 T^{2} + 21 p T^{3} + 111 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 43 T^{2} - 185 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 67 T^{2} + 584 T^{3} + 67 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 2 T + 59 T^{2} - 68 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 149 T^{2} - 928 T^{3} + 149 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 4 T + 109 T^{2} + 312 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 5 T + 123 T^{2} + 573 T^{3} + 123 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 5 T + 141 T^{2} - 423 T^{3} + 141 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 22 T + 335 T^{2} - 3012 T^{3} + 335 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 185 T^{2} - 700 T^{3} + 185 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 18 T + 293 T^{2} - 2548 T^{3} + 293 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 13 T + 189 T^{2} + 1885 T^{3} + 189 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 13 T + 121 T^{2} + 591 T^{3} + 121 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 14 T + 211 T^{2} - 2436 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 23 T + 381 T^{2} + 47 p T^{3} + 381 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.88236551529430493639098124491, −7.61957376177987217980668224026, −7.53348813730961068032993068489, −7.25508119855126045534293106889, −6.91326395128430696223958849100, −6.57342157736446563203640641089, −6.45273075658842531758478673511, −5.77908030267344464883544330473, −5.67481891642127170646235318343, −5.35976740807033976007140857700, −5.14245254158200569650425910312, −5.04938244221816835769405207861, −4.82555507285717759901511031324, −4.08922575539576788886158176819, −3.94604409854709923067537256937, −3.86170787364741677467688368514, −3.16138156393720484867705249829, −2.91143786590627874530416737651, −2.50475291695567234081807848223, −2.08695246348184295964733311291, −1.91637658513295284154592585370, −1.72519437429064030291952691679, −1.19961268158426854126058495289, −0.837956211942937266197640613796, −0.46080461336512764840134095766,
0.46080461336512764840134095766, 0.837956211942937266197640613796, 1.19961268158426854126058495289, 1.72519437429064030291952691679, 1.91637658513295284154592585370, 2.08695246348184295964733311291, 2.50475291695567234081807848223, 2.91143786590627874530416737651, 3.16138156393720484867705249829, 3.86170787364741677467688368514, 3.94604409854709923067537256937, 4.08922575539576788886158176819, 4.82555507285717759901511031324, 5.04938244221816835769405207861, 5.14245254158200569650425910312, 5.35976740807033976007140857700, 5.67481891642127170646235318343, 5.77908030267344464883544330473, 6.45273075658842531758478673511, 6.57342157736446563203640641089, 6.91326395128430696223958849100, 7.25508119855126045534293106889, 7.53348813730961068032993068489, 7.61957376177987217980668224026, 7.88236551529430493639098124491
