| 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 − 6·11-s + 18·12-s − 6·13-s − 9·14-s − 9·15-s + 15·16-s + 18·18-s − 6·19-s − 18·20-s − 9·21-s − 18·22-s − 3·23-s + 30·24-s + 6·25-s − 18·26-s + 10·27-s − 18·28-s − 27·30-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 − 1.80·11-s + 5.19·12-s − 1.66·13-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 − 3.83·22-s − 0.625·23-s + 6.12·24-s + 6/5·25-s − 3.53·26-s + 1.92·27-s − 3.40·28-s − 4.92·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 11 | $A_4\times C_2$ | \( 1 + 6 T + 3 p T^{2} + 108 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 6 T + 15 T^{2} + 20 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 15 T^{2} - 72 T^{3} + 15 p T^{4} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T - 15 T^{2} - 12 p T^{3} - 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T^{2} + 136 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 18 T + 189 T^{2} + 1252 T^{3} + 189 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 420 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 468 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 18 T + 201 T^{2} + 1476 T^{3} + 201 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 18 T + 237 T^{2} + 1828 T^{3} + 237 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 219 T^{2} - 1900 T^{3} + 219 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 147 T^{2} - 580 T^{3} + 147 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 225 T^{2} + 1988 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 428 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 39 T^{2} + 12 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 93 T^{2} + 576 T^{3} + 93 p T^{4} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 18 T + 3 p T^{2} + 2772 T^{3} + 3 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 6 T + 27 T^{2} + 644 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 18 T + 243 T^{2} + 2620 T^{3} + 243 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.73268376502656463979251646501, −7.21652938233587827207326027774, −7.20370763161807163535741180210, −7.14242083433792850185529341978, −6.62563118135623564516610298340, −6.59518440476841374644516717990, −6.40131603951504642406281611021, −5.78321770031843253890311907476, −5.52947691115533599303161619472, −5.47260361813499900312505240209, −4.97923424993422224167996869478, −4.93731609297686679087122294637, −4.74666863785233797986517879415, −4.18031290256600266143534772887, −4.15971500023953936868576923332, −3.85897896507402793945061928095, −3.60690133890794715064177013249, −3.31864085868036110377219379685, −3.13152754476590299240531311096, −2.80494197108061342409356203951, −2.64980081967151006430534305117, −2.50784287698396036699148366748, −1.74707860790816950376884425499, −1.69710551190677417711755627390, −1.66716924592245928289153577210, 0, 0, 0,
1.66716924592245928289153577210, 1.69710551190677417711755627390, 1.74707860790816950376884425499, 2.50784287698396036699148366748, 2.64980081967151006430534305117, 2.80494197108061342409356203951, 3.13152754476590299240531311096, 3.31864085868036110377219379685, 3.60690133890794715064177013249, 3.85897896507402793945061928095, 4.15971500023953936868576923332, 4.18031290256600266143534772887, 4.74666863785233797986517879415, 4.93731609297686679087122294637, 4.97923424993422224167996869478, 5.47260361813499900312505240209, 5.52947691115533599303161619472, 5.78321770031843253890311907476, 6.40131603951504642406281611021, 6.59518440476841374644516717990, 6.62563118135623564516610298340, 7.14242083433792850185529341978, 7.20370763161807163535741180210, 7.21652938233587827207326027774, 7.73268376502656463979251646501
