| L(s) = 1 | + 2-s + 3·3-s − 3·4-s − 3·5-s + 3·6-s + 3·7-s − 4·8-s + 6·9-s − 3·10-s − 11-s − 9·12-s + 3·14-s − 9·15-s + 3·16-s − 7·17-s + 6·18-s − 3·19-s + 9·20-s + 9·21-s − 22-s − 12·23-s − 12·24-s − 9·25-s + 10·27-s − 9·28-s − 29-s − 9·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 3/2·4-s − 1.34·5-s + 1.22·6-s + 1.13·7-s − 1.41·8-s + 2·9-s − 0.948·10-s − 0.301·11-s − 2.59·12-s + 0.801·14-s − 2.32·15-s + 3/4·16-s − 1.69·17-s + 1.41·18-s − 0.688·19-s + 2.01·20-s + 1.96·21-s − 0.213·22-s − 2.50·23-s − 2.44·24-s − 9/5·25-s + 1.92·27-s − 1.70·28-s − 0.185·29-s − 1.64·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - T + p^{2} T^{2} - 3 T^{3} + p^{3} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 31 T^{2} + 21 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 39 T^{2} + 101 T^{3} + 39 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 12 T + 110 T^{2} + 595 T^{3} + 110 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + T + 71 T^{2} + p T^{3} + 71 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - T + 91 T^{2} - 61 T^{3} + 91 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 7 T + 90 T^{2} + 427 T^{3} + 90 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 8 T + 100 T^{2} - 669 T^{3} + 100 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 47 | $A_4\times C_2$ | \( 1 - 12 T + 126 T^{2} - 751 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - T + 38 T^{2} + 343 T^{3} + 38 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 21 T + 296 T^{2} - 2569 T^{3} + 296 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 20 T + 300 T^{2} + 2609 T^{3} + 300 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 228 T^{2} - 1581 T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 204 T^{2} + 825 T^{3} + 204 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 3 T + 201 T^{2} + 425 T^{3} + 201 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 173 T^{2} + 1463 T^{3} + 173 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 2 T + 150 T^{2} + 345 T^{3} + 150 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 19 T + 322 T^{2} + 3411 T^{3} + 322 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 16 T + 290 T^{2} + 2487 T^{3} + 290 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.158613268467312866010030572431, −7.71102779761838577077348011044, −7.61290675687922859644606532921, −7.47262399111939498077102813850, −7.03388247860469795216391390893, −6.84871285854619955189631565162, −6.48529011540865722048155750099, −6.14675147449184485710798883392, −5.85518154487398348206560196640, −5.64885582003639708585284987656, −5.14758981996351563025383767815, −4.95739781800584039905924149328, −4.88968810062568836162220365077, −4.32233990711511806864696962404, −4.12631422060932095175677673043, −4.11538371912158495722877716128, −3.85194724271768377235833383235, −3.72298479040942644822523626151, −3.53045646943715648794044420960, −2.71432169642586701753418947213, −2.64780194766641664650203894302, −2.26971061501100233011248395891, −2.01186929585159650720688141807, −1.48268808518074553761528145771, −1.35827051610252655844435914678, 0, 0, 0,
1.35827051610252655844435914678, 1.48268808518074553761528145771, 2.01186929585159650720688141807, 2.26971061501100233011248395891, 2.64780194766641664650203894302, 2.71432169642586701753418947213, 3.53045646943715648794044420960, 3.72298479040942644822523626151, 3.85194724271768377235833383235, 4.11538371912158495722877716128, 4.12631422060932095175677673043, 4.32233990711511806864696962404, 4.88968810062568836162220365077, 4.95739781800584039905924149328, 5.14758981996351563025383767815, 5.64885582003639708585284987656, 5.85518154487398348206560196640, 6.14675147449184485710798883392, 6.48529011540865722048155750099, 6.84871285854619955189631565162, 7.03388247860469795216391390893, 7.47262399111939498077102813850, 7.61290675687922859644606532921, 7.71102779761838577077348011044, 8.158613268467312866010030572431
