| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 8·5-s + 9·6-s + 3·7-s + 10·8-s + 6·9-s − 24·10-s − 5·11-s + 18·12-s + 9·14-s − 24·15-s + 15·16-s − 3·17-s + 18·18-s − 5·19-s − 48·20-s + 9·21-s − 15·22-s − 7·23-s + 30·24-s + 30·25-s + 10·27-s + 18·28-s − 4·29-s − 72·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 3.57·5-s + 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s − 7.58·10-s − 1.50·11-s + 5.19·12-s + 2.40·14-s − 6.19·15-s + 15/4·16-s − 0.727·17-s + 4.24·18-s − 1.14·19-s − 10.7·20-s + 1.96·21-s − 3.19·22-s − 1.45·23-s + 6.12·24-s + 6·25-s + 1.92·27-s + 3.40·28-s − 0.742·29-s − 13.1·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 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 | $A_4\times C_2$ | \( 1 + 5 T + 39 T^{2} + 111 T^{3} + 39 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 89 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 5 T + 49 T^{2} + 149 T^{3} + 49 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 7 T + 55 T^{2} + 315 T^{3} + 55 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 4 T + 62 T^{2} + 161 T^{3} + 62 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 4 T + 26 T^{2} - 219 T^{3} + 26 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 2 T + 82 T^{2} - 77 T^{3} + 82 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 18 T + 168 T^{2} + 1125 T^{3} + 168 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 121 T^{2} - 389 T^{3} + 121 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 23 T + 308 T^{2} + 2539 T^{3} + 308 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 12 T + 116 T^{2} + 1175 T^{3} + 116 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1425 T^{3} + 162 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 17 T + 214 T^{2} - 2033 T^{3} + 214 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 10 T + 162 T^{2} + 991 T^{3} + 162 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 14 T + 241 T^{2} - 1932 T^{3} + 241 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 23 T + 379 T^{2} + 3707 T^{3} + 379 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 4 T + 170 T^{2} - 603 T^{3} + 170 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 37 T + 689 T^{2} + 7835 T^{3} + 689 p T^{4} + 37 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 10 T + 186 T^{2} - 1893 T^{3} + 186 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 20 T + 338 T^{2} + 3909 T^{3} + 338 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.59661896356100301442237919312, −7.15465037313848863975219729532, −6.93591323045714733440255940818, −6.82377829606075238733671206257, −6.40041484120188797609936776030, −6.28237378449982462111439412594, −5.99839664684867970292191146167, −5.41559323758850949349930480095, −5.32927300650717325661312646451, −5.02036399523995333640704698599, −4.71730452925961442318067791754, −4.57392320744024706954975973969, −4.38491301498735226811210107398, −4.24741445824610656116778087691, −3.89508149228503824607364193601, −3.83231141584474201706806802508, −3.42366586246942083503164432799, −3.33981023526476858745438183164, −3.15470782148629430912775964496, −2.57479431814433403163452977269, −2.49260533850567831785645397877, −2.42477940547994516899024362149, −1.63896154928425624085374196495, −1.52045292117815995442945164803, −1.40701266521261059854509410140, 0, 0, 0,
1.40701266521261059854509410140, 1.52045292117815995442945164803, 1.63896154928425624085374196495, 2.42477940547994516899024362149, 2.49260533850567831785645397877, 2.57479431814433403163452977269, 3.15470782148629430912775964496, 3.33981023526476858745438183164, 3.42366586246942083503164432799, 3.83231141584474201706806802508, 3.89508149228503824607364193601, 4.24741445824610656116778087691, 4.38491301498735226811210107398, 4.57392320744024706954975973969, 4.71730452925961442318067791754, 5.02036399523995333640704698599, 5.32927300650717325661312646451, 5.41559323758850949349930480095, 5.99839664684867970292191146167, 6.28237378449982462111439412594, 6.40041484120188797609936776030, 6.82377829606075238733671206257, 6.93591323045714733440255940818, 7.15465037313848863975219729532, 7.59661896356100301442237919312
