| L(s) = 1 | + 2·2-s − 3·3-s + 4-s + 3·5-s − 6·6-s + 3·7-s + 2·8-s + 6·9-s + 6·10-s + 2·11-s − 3·12-s + 6·14-s − 9·15-s + 7·16-s − 8·17-s + 12·18-s + 7·19-s + 3·20-s − 9·21-s + 4·22-s − 9·23-s − 6·24-s − 2·25-s − 10·27-s + 3·28-s − 29-s − 18·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 1/2·4-s + 1.34·5-s − 2.44·6-s + 1.13·7-s + 0.707·8-s + 2·9-s + 1.89·10-s + 0.603·11-s − 0.866·12-s + 1.60·14-s − 2.32·15-s + 7/4·16-s − 1.94·17-s + 2.82·18-s + 1.60·19-s + 0.670·20-s − 1.96·21-s + 0.852·22-s − 1.87·23-s − 1.22·24-s − 2/5·25-s − 1.92·27-s + 0.566·28-s − 0.185·29-s − 3.28·30-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
| \(L(1)\) |
\(\approx\) |
\(8.964039385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.964039385\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{6}$ | \( 1 - p T + 3 T^{2} - 3 p T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 22 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $D_{6}$ | \( 1 - 2 T + 5 T^{2} - 52 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 55 T^{2} + 240 T^{3} + 55 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 41 T^{2} - 138 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 422 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 55 T^{2} - 18 T^{3} + 55 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 7 T + 53 T^{2} - 162 T^{3} + 53 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 131 T^{2} + 856 T^{3} + 131 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 139 T^{2} - 804 T^{3} + 139 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 113 T^{2} + 102 T^{3} + 113 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 17 T + 221 T^{2} - 1666 T^{3} + 221 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 5 T + 63 T^{2} + 678 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 197 T^{2} - 1352 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 147 T^{2} - 988 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 73 T^{2} - 340 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 121 T^{2} - 72 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 5 T + 75 T^{2} - 1166 T^{3} + 75 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 13 T + 277 T^{2} + 2086 T^{3} + 277 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - T + 217 T^{2} - 90 T^{3} + 217 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 13 T + 263 T^{2} - 1970 T^{3} + 263 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 107 T^{2} - 1222 T^{3} + 107 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.57107563960930753519863009185, −7.11075390237663780609044862465, −6.88482338249029138918511470326, −6.78788834237554323647338468499, −6.51835484969991427695785918959, −6.15033881035853660856639123554, −5.81376749217112392053175894260, −5.62569283122265953194289408590, −5.56875144432299329893235459045, −5.52797590333693566233282722083, −4.91936280999292724020353589427, −4.80304128667275839724889850078, −4.61660166986362019261862130228, −4.24702301754361313654741081306, −4.11759481753334824398912989263, −3.89343117721360334667801375739, −3.61359690317342589812879793717, −3.13414307907434150296015994632, −2.47021373364816756057608057587, −2.35747902867120022967162708923, −1.83552203706708239277823911980, −1.82607774419445572838067052567, −1.37719267456122149099576943674, −0.74302106059327389029025927391, −0.63914304032631705766319181516,
0.63914304032631705766319181516, 0.74302106059327389029025927391, 1.37719267456122149099576943674, 1.82607774419445572838067052567, 1.83552203706708239277823911980, 2.35747902867120022967162708923, 2.47021373364816756057608057587, 3.13414307907434150296015994632, 3.61359690317342589812879793717, 3.89343117721360334667801375739, 4.11759481753334824398912989263, 4.24702301754361313654741081306, 4.61660166986362019261862130228, 4.80304128667275839724889850078, 4.91936280999292724020353589427, 5.52797590333693566233282722083, 5.56875144432299329893235459045, 5.62569283122265953194289408590, 5.81376749217112392053175894260, 6.15033881035853660856639123554, 6.51835484969991427695785918959, 6.78788834237554323647338468499, 6.88482338249029138918511470326, 7.11075390237663780609044862465, 7.57107563960930753519863009185
