| L(s) = 1 | + 3·3-s + 3·7-s + 6·9-s − 6·11-s − 6·13-s − 6·19-s + 9·21-s − 4·23-s + 10·27-s + 2·29-s − 2·31-s − 18·33-s − 4·37-s − 18·39-s + 2·41-s − 4·43-s − 8·47-s + 6·49-s + 14·53-s − 18·57-s − 16·59-s − 6·61-s + 18·63-s − 8·67-s − 12·69-s − 6·71-s − 18·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.13·7-s + 2·9-s − 1.80·11-s − 1.66·13-s − 1.37·19-s + 1.96·21-s − 0.834·23-s + 1.92·27-s + 0.371·29-s − 0.359·31-s − 3.13·33-s − 0.657·37-s − 2.88·39-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 6/7·49-s + 1.92·53-s − 2.38·57-s − 2.08·59-s − 0.768·61-s + 2.26·63-s − 0.977·67-s − 1.44·69-s − 0.712·71-s − 2.10·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 31 T^{2} + 232 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T - 15 T^{2} - 488 T^{3} - 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 171 T^{2} - 1188 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 18 T + 311 T^{2} + 2732 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 255 T^{2} + 2404 T^{3} + 255 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.47934594478428901313514716874, −7.08722493499518440738328498494, −6.87074677222050492032592965002, −6.81872570677881479675696407270, −6.12771485872773155172176225113, −6.10666355790468542098737697768, −6.00916801892294086881475503877, −5.39439111564002220496709732544, −5.23331604848420624072848315191, −5.14174151199499004874680761153, −4.83606483509197701024339481103, −4.65095523135277921817141685338, −4.29132123597451783806593681388, −4.16966700031446815108481058914, −3.86511510760821681442361949879, −3.77473157744390006564836304931, −2.99663493002885505476199542971, −2.99309322700276054988869272948, −2.89411502326840353820667977345, −2.32197561534704457594410000525, −2.32050792731263450448867675621, −2.27393248189696305289675508042, −1.42653788348222441197506851037, −1.42054532608382839771610746360, −1.39955950727859306225437995968, 0, 0, 0,
1.39955950727859306225437995968, 1.42054532608382839771610746360, 1.42653788348222441197506851037, 2.27393248189696305289675508042, 2.32050792731263450448867675621, 2.32197561534704457594410000525, 2.89411502326840353820667977345, 2.99309322700276054988869272948, 2.99663493002885505476199542971, 3.77473157744390006564836304931, 3.86511510760821681442361949879, 4.16966700031446815108481058914, 4.29132123597451783806593681388, 4.65095523135277921817141685338, 4.83606483509197701024339481103, 5.14174151199499004874680761153, 5.23331604848420624072848315191, 5.39439111564002220496709732544, 6.00916801892294086881475503877, 6.10666355790468542098737697768, 6.12771485872773155172176225113, 6.81872570677881479675696407270, 6.87074677222050492032592965002, 7.08722493499518440738328498494, 7.47934594478428901313514716874
