| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s + 6·5-s − 9·6-s + 3·7-s − 10·8-s + 6·9-s − 18·10-s + 6·11-s + 18·12-s + 6·13-s − 9·14-s + 18·15-s + 15·16-s − 18·18-s − 6·19-s + 36·20-s + 9·21-s − 18·22-s + 18·23-s − 30·24-s + 12·25-s − 18·26-s + 10·27-s + 18·28-s − 54·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s + 2.68·5-s − 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s − 5.69·10-s + 1.80·11-s + 5.19·12-s + 1.66·13-s − 2.40·14-s + 4.64·15-s + 15/4·16-s − 4.24·18-s − 1.37·19-s + 8.04·20-s + 1.96·21-s − 3.83·22-s + 3.75·23-s − 6.12·24-s + 12/5·25-s − 3.53·26-s + 1.92·27-s + 3.40·28-s − 9.85·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.769915924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.769915924\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 17 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 23 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 81 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 132 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 92 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 29 | $A_4\times C_2$ | \( 1 + 30 T^{2} - 163 T^{3} + 30 p T^{4} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 6 T + 84 T^{2} + 375 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 39 T^{2} - 4 p T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 628 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 6 T - 3 T^{2} + 340 T^{3} - 3 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 57 T^{2} + 296 T^{3} + 57 p T^{4} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 565 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 12 T + 222 T^{2} - 1467 T^{3} + 222 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 30 T + 471 T^{2} + 4532 T^{3} + 471 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1200 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T - 3 T^{2} - 900 T^{3} - 3 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 24 T + 384 T^{2} + 3773 T^{3} + 384 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 90 T^{2} - 683 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 15 T + 240 T^{2} + 2331 T^{3} + 240 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 24 T + 303 T^{2} - 2824 T^{3} + 303 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 12 T + 228 T^{2} + 1515 T^{3} + 228 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.496480637337237160766735168238, −8.183710451733770839657096656056, −7.85130118096693128902012009223, −7.61458522149161152199980686007, −7.31261737332279640263247102368, −6.94094267493458114138476821819, −6.87204783083030700086693436650, −6.40152267599825505795736649924, −6.29934184356783609710143746867, −6.04287059840762839447030137145, −5.73561344770908146518248049510, −5.28834859167449408234517948815, −5.22517462281354628851040665189, −4.39512864114406906844855877196, −4.17898172373657558340596551340, −4.15012373346960019680499265936, −3.29387469437886062356139845421, −3.09993793259542840564766989603, −2.89972227691603271327347879321, −2.34686318863582020257373824856, −2.01870138418248886970439854072, −1.83826725377583472285432369993, −1.43219672641678968739743428051, −1.13605763501870202493173734514, −1.07163750474932699587804430994,
1.07163750474932699587804430994, 1.13605763501870202493173734514, 1.43219672641678968739743428051, 1.83826725377583472285432369993, 2.01870138418248886970439854072, 2.34686318863582020257373824856, 2.89972227691603271327347879321, 3.09993793259542840564766989603, 3.29387469437886062356139845421, 4.15012373346960019680499265936, 4.17898172373657558340596551340, 4.39512864114406906844855877196, 5.22517462281354628851040665189, 5.28834859167449408234517948815, 5.73561344770908146518248049510, 6.04287059840762839447030137145, 6.29934184356783609710143746867, 6.40152267599825505795736649924, 6.87204783083030700086693436650, 6.94094267493458114138476821819, 7.31261737332279640263247102368, 7.61458522149161152199980686007, 7.85130118096693128902012009223, 8.183710451733770839657096656056, 8.496480637337237160766735168238
