| L(s) = 1 | + 3·2-s + 6·4-s + 3·5-s − 6·7-s + 10·8-s + 9·10-s + 3·11-s − 18·14-s + 15·16-s + 9·17-s − 3·19-s + 18·20-s + 9·22-s + 15·23-s + 3·25-s − 36·28-s − 6·29-s − 3·31-s + 21·32-s + 27·34-s − 18·35-s − 6·37-s − 9·38-s + 30·40-s + 9·41-s − 21·43-s + 18·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 1.34·5-s − 2.26·7-s + 3.53·8-s + 2.84·10-s + 0.904·11-s − 4.81·14-s + 15/4·16-s + 2.18·17-s − 0.688·19-s + 4.02·20-s + 1.91·22-s + 3.12·23-s + 3/5·25-s − 6.80·28-s − 1.11·29-s − 0.538·31-s + 3.71·32-s + 4.63·34-s − 3.04·35-s − 0.986·37-s − 1.45·38-s + 4.74·40-s + 1.40·41-s − 3.20·43-s + 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.59251081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.59251081\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 12 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} + 27 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 69 T^{2} - 302 T^{3} + 69 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 15 T + 135 T^{2} - 766 T^{3} + 135 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 201 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 42 T^{2} + 52 T^{3} + 42 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 348 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 9 T + 78 T^{2} - 616 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 222 T^{2} + 1622 T^{3} + 222 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 81 T^{2} - 456 T^{3} + 81 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 69 T^{2} - 502 T^{3} + 69 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 153 T^{2} - 290 T^{3} + 153 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 24 T + 339 T^{2} - 3120 T^{3} + 339 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 105 T^{2} + 123 T^{3} + 105 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T + 216 T^{2} + 1676 T^{3} + 216 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 213 T^{2} + 426 T^{3} + 213 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 4 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 33 T + 594 T^{2} - 6582 T^{3} + 594 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 231 T^{2} + 924 T^{3} + 231 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 195 T^{2} - 2072 T^{3} + 195 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.42287122894200064408034787332, −7.01331555380641836309007651757, −6.88720221893020516072447734127, −6.73205219702496126636342677607, −6.44924272599378475146283652795, −6.19833382264259948326580291927, −5.99468198560774501128534429134, −5.61780377378520884971941482264, −5.52616891795948409511824633647, −5.47714578365147635211939259268, −4.94620382366068598508525214525, −4.87477921070852655905321057952, −4.55211498613396365470103615904, −3.81638242429452264299338139202, −3.76898583652084982082407561951, −3.76209089006245071027610009514, −3.26004313425429930875772700942, −3.19403288764435951052283397951, −2.87028323279788616978177319180, −2.38068652390596721892413141053, −2.32825021569753419400491368980, −1.83438081712256438813287935059, −1.33253935861584840610752798828, −1.05988812158155640401461105704, −0.57792624091526970490551875298,
0.57792624091526970490551875298, 1.05988812158155640401461105704, 1.33253935861584840610752798828, 1.83438081712256438813287935059, 2.32825021569753419400491368980, 2.38068652390596721892413141053, 2.87028323279788616978177319180, 3.19403288764435951052283397951, 3.26004313425429930875772700942, 3.76209089006245071027610009514, 3.76898583652084982082407561951, 3.81638242429452264299338139202, 4.55211498613396365470103615904, 4.87477921070852655905321057952, 4.94620382366068598508525214525, 5.47714578365147635211939259268, 5.52616891795948409511824633647, 5.61780377378520884971941482264, 5.99468198560774501128534429134, 6.19833382264259948326580291927, 6.44924272599378475146283652795, 6.73205219702496126636342677607, 6.88720221893020516072447734127, 7.01331555380641836309007651757, 7.42287122894200064408034787332
