| L(s) = 1 | − 3·3-s − 3·7-s + 3·9-s + 3·11-s − 3·13-s − 3·17-s − 3·19-s + 9·21-s − 6·23-s − 3·27-s + 12·29-s + 3·31-s − 9·33-s + 12·37-s + 9·39-s + 6·41-s + 3·43-s − 12·47-s − 3·49-s + 9·51-s − 9·53-s + 9·57-s − 24·59-s − 3·61-s − 9·63-s − 6·67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 9-s + 0.904·11-s − 0.832·13-s − 0.727·17-s − 0.688·19-s + 1.96·21-s − 1.25·23-s − 0.577·27-s + 2.22·29-s + 0.538·31-s − 1.56·33-s + 1.97·37-s + 1.44·39-s + 0.937·41-s + 0.457·43-s − 1.75·47-s − 3/7·49-s + 1.26·51-s − 1.23·53-s + 1.19·57-s − 3.12·59-s − 0.384·61-s − 1.13·63-s − 0.733·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 2 p T^{2} + 4 p T^{3} + 2 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 12 T^{2} + 46 T^{3} + 12 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 63 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} - 42 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 271 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 123 T^{2} - 727 T^{3} + 123 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 48 T^{2} - 259 T^{3} + 48 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 138 T^{2} - 884 T^{3} + 138 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 120 T^{2} - 472 T^{3} + 120 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 117 T^{2} - 246 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 1120 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 90 T^{2} + 412 T^{3} + 90 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 24 T + 312 T^{2} + 2732 T^{3} + 312 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 72 T^{2} + 20 T^{3} + 72 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 153 T^{2} + 676 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 15 T + 111 T^{2} - 630 T^{3} + 111 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 33 T + 570 T^{2} + 6002 T^{3} + 570 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 15 T + 258 T^{2} + 2198 T^{3} + 258 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 147 T^{2} + 1009 T^{3} + 147 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 165 T^{2} - 615 T^{3} + 165 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 18 T + 171 T^{2} + 1015 T^{3} + 171 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.77749848156350532630341386479, −7.66328718696255154077118126343, −7.02592286658170180911874833097, −6.99183723167994755540760901583, −6.62340532059174174283073041674, −6.35986554270084087487157755702, −6.19363513780675348860031476919, −6.12076013953186846797784659814, −6.01172807671707852095711789840, −5.80840134854033487356741103098, −5.18825381742332467794144550096, −5.01686166503941793220868552915, −4.78725949116509166752919833130, −4.43161215763664818660334580906, −4.33239291914919662305595248509, −4.19981592824679162461799832749, −3.76430628065663320258191892349, −3.16319403514442045040955733300, −3.14849243987786485995698908944, −2.80334525384440957543101206241, −2.50810951274643824668381144977, −2.09209916746259551620164255363, −1.68214509556872672569426857033, −1.13993894243470798599645200060, −1.07289021345290563485646633503, 0, 0, 0,
1.07289021345290563485646633503, 1.13993894243470798599645200060, 1.68214509556872672569426857033, 2.09209916746259551620164255363, 2.50810951274643824668381144977, 2.80334525384440957543101206241, 3.14849243987786485995698908944, 3.16319403514442045040955733300, 3.76430628065663320258191892349, 4.19981592824679162461799832749, 4.33239291914919662305595248509, 4.43161215763664818660334580906, 4.78725949116509166752919833130, 5.01686166503941793220868552915, 5.18825381742332467794144550096, 5.80840134854033487356741103098, 6.01172807671707852095711789840, 6.12076013953186846797784659814, 6.19363513780675348860031476919, 6.35986554270084087487157755702, 6.62340532059174174283073041674, 6.99183723167994755540760901583, 7.02592286658170180911874833097, 7.66328718696255154077118126343, 7.77749848156350532630341386479
