| L(s) = 1 | − 3·2-s + 6·4-s + 3·5-s + 3·7-s − 10·8-s − 9·10-s − 11-s − 9·14-s + 15·16-s − 12·17-s + 4·19-s + 18·20-s + 3·22-s − 16·23-s − 2·25-s + 18·28-s − 13·29-s − 9·31-s − 21·32-s + 36·34-s + 9·35-s − 12·37-s − 12·38-s − 30·40-s + 14·41-s − 8·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.34·5-s + 1.13·7-s − 3.53·8-s − 2.84·10-s − 0.301·11-s − 2.40·14-s + 15/4·16-s − 2.91·17-s + 0.917·19-s + 4.02·20-s + 0.639·22-s − 3.33·23-s − 2/5·25-s + 3.40·28-s − 2.41·29-s − 1.61·31-s − 3.71·32-s + 6.17·34-s + 1.52·35-s − 1.97·37-s − 1.94·38-s − 4.74·40-s + 2.18·41-s − 1.21·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 31 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 17 T^{2} - 43 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 17 T^{2} + 35 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 12 T + 71 T^{2} + 304 T^{3} + 71 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 4 T + 25 T^{2} - 88 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 16 T + 145 T^{2} + 840 T^{3} + 145 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 13 T + 99 T^{2} + 531 T^{3} + 99 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 9 T + 71 T^{2} + 529 T^{3} + 71 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 12 T + 131 T^{2} + 896 T^{3} + 131 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 14 T + 179 T^{2} - 1204 T^{3} + 179 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 8 T + 85 T^{2} + 8 p T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 4 T + 109 T^{2} + 312 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 15 T + 87 T^{2} + 343 T^{3} + 87 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 9 T + 197 T^{2} - 1075 T^{3} + 197 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 10 T + 207 T^{2} + 1228 T^{3} + 207 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 308 T^{3} + 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 141 T^{2} + 748 T^{3} + 141 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 5 T + 197 T^{2} - 717 T^{3} + 197 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 5 T + 33 T^{2} - 679 T^{3} + 33 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 7 T + 25 T^{2} + 315 T^{3} + 25 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 10 T + 291 T^{2} - 1788 T^{3} + 291 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 7 T + 277 T^{2} - 1351 T^{3} + 277 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.131684930390957908656536105406, −7.88517396292165238930093406755, −7.73971639737546669005969671589, −7.54494654803559069286065657438, −7.11669937392378610449011238461, −7.05483972315288101485203920033, −6.57250502510564158718889110132, −6.34713791665912909635975738324, −6.20396114467422436656260008460, −6.01565439912088904319511871196, −5.43320811743639688314070191549, −5.42805428604844832595547951041, −5.35874962901306720277290434931, −4.73466101107536402627638449653, −4.31627354366123087736257362815, −4.24841263267134275871458242060, −3.65713551921664261851533644599, −3.56251804291097970523003834742, −3.11140461696684152414229176295, −2.40420703742196743148810854159, −2.32267504672414293116200651629, −2.05880920347674677370513617734, −1.68121565781626644472996235908, −1.67729847054278109645170152862, −1.38923057390762519928632329771, 0, 0, 0,
1.38923057390762519928632329771, 1.67729847054278109645170152862, 1.68121565781626644472996235908, 2.05880920347674677370513617734, 2.32267504672414293116200651629, 2.40420703742196743148810854159, 3.11140461696684152414229176295, 3.56251804291097970523003834742, 3.65713551921664261851533644599, 4.24841263267134275871458242060, 4.31627354366123087736257362815, 4.73466101107536402627638449653, 5.35874962901306720277290434931, 5.42805428604844832595547951041, 5.43320811743639688314070191549, 6.01565439912088904319511871196, 6.20396114467422436656260008460, 6.34713791665912909635975738324, 6.57250502510564158718889110132, 7.05483972315288101485203920033, 7.11669937392378610449011238461, 7.54494654803559069286065657438, 7.73971639737546669005969671589, 7.88517396292165238930093406755, 8.131684930390957908656536105406
