| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s + 3·5-s − 9·6-s + 3·7-s + 10·8-s + 6·9-s + 9·10-s + 2·11-s − 18·12-s + 2·13-s + 9·14-s − 9·15-s + 15·16-s + 18·18-s + 6·19-s + 18·20-s − 9·21-s + 6·22-s − 3·23-s − 30·24-s + 6·25-s + 6·26-s − 10·27-s + 18·28-s + 4·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s + 2.84·10-s + 0.603·11-s − 5.19·12-s + 0.554·13-s + 2.40·14-s − 2.32·15-s + 15/4·16-s + 4.24·18-s + 1.37·19-s + 4.02·20-s − 1.96·21-s + 1.27·22-s − 0.625·23-s − 6.12·24-s + 6/5·25-s + 1.17·26-s − 1.92·27-s + 3.40·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(30.63705021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(30.63705021\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 60 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 20 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 8 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 212 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 71 T^{2} - 176 T^{3} + 71 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 340 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 59 T^{2} - 260 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 71 T^{2} + 436 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 22 T + 269 T^{2} - 2100 T^{3} + 269 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 532 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 107 T^{2} - 452 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 113 T^{2} - 64 T^{3} + 113 p T^{4} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 181 T^{2} + 284 T^{3} + 181 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 33 T^{2} + 332 T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 183 T^{2} - 1796 T^{3} + 183 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 924 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 283 T^{2} + 2268 T^{3} + 283 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 91 T^{2} - 796 T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.25311558767002929763351719735, −6.68912461121706908581559208455, −6.62397861365678179187300773994, −6.62083898252875593960871916940, −6.05569991959587379452565212302, −5.96278981400793969504747030319, −5.89016291864801246473997044149, −5.57738678628620497288918642812, −5.27659430406985619739790900959, −5.23573169650254509806830616711, −4.93916748688378984757777106575, −4.61443574394053645775748783633, −4.38580496571612401553306097828, −4.10178960771677848846864564130, −4.02652014403058145512949721579, −3.73178600669656581546052768675, −3.05973110777800318461065969525, −3.01948285864853911944164422164, −2.62961668710100263802782099444, −2.24219600718194982659014658916, −1.94348553472040814541728902041, −1.71514378027106921447602565493, −1.13810353540485618513726464980, −0.935834192190042340345114247002, −0.816445402049001905717273919158,
0.816445402049001905717273919158, 0.935834192190042340345114247002, 1.13810353540485618513726464980, 1.71514378027106921447602565493, 1.94348553472040814541728902041, 2.24219600718194982659014658916, 2.62961668710100263802782099444, 3.01948285864853911944164422164, 3.05973110777800318461065969525, 3.73178600669656581546052768675, 4.02652014403058145512949721579, 4.10178960771677848846864564130, 4.38580496571612401553306097828, 4.61443574394053645775748783633, 4.93916748688378984757777106575, 5.23573169650254509806830616711, 5.27659430406985619739790900959, 5.57738678628620497288918642812, 5.89016291864801246473997044149, 5.96278981400793969504747030319, 6.05569991959587379452565212302, 6.62083898252875593960871916940, 6.62397861365678179187300773994, 6.68912461121706908581559208455, 7.25311558767002929763351719735
