| L(s) = 1 | + 6·2-s − 9·3-s + 24·4-s + 7·5-s − 54·6-s − 21·7-s + 80·8-s + 54·9-s + 42·10-s − 47·11-s − 216·12-s + 39·13-s − 126·14-s − 63·15-s + 240·16-s + 119·17-s + 324·18-s + 101·19-s + 168·20-s + 189·21-s − 282·22-s − 27·23-s − 720·24-s − 30·25-s + 234·26-s − 270·27-s − 504·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 0.626·5-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 1.32·10-s − 1.28·11-s − 5.19·12-s + 0.832·13-s − 2.40·14-s − 1.08·15-s + 15/4·16-s + 1.69·17-s + 4.24·18-s + 1.21·19-s + 1.87·20-s + 1.96·21-s − 2.73·22-s − 0.244·23-s − 6.12·24-s − 0.239·25-s + 1.76·26-s − 1.92·27-s − 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.03427921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.03427921\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 7 T + 79 T^{2} + 214 T^{3} + 79 p^{3} T^{4} - 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 47 T + 3009 T^{2} + 121666 T^{3} + 3009 p^{3} T^{4} + 47 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 p T + 12491 T^{2} - 975258 T^{3} + 12491 p^{3} T^{4} - 7 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 101 T + 14313 T^{2} - 758110 T^{3} + 14313 p^{3} T^{4} - 101 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 27 T + 8509 T^{2} + 1749194 T^{3} + 8509 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 345 T + 93951 T^{2} - 16062286 T^{3} + 93951 p^{3} T^{4} - 345 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 152 T + 39869 T^{2} - 2696656 T^{3} + 39869 p^{3} T^{4} - 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 227 T + 60543 T^{2} - 8893578 T^{3} + 60543 p^{3} T^{4} - 227 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 452 T + 165895 T^{2} - 40174120 T^{3} + 165895 p^{3} T^{4} - 452 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 415 T + 193329 T^{2} + 55010826 T^{3} + 193329 p^{3} T^{4} + 415 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 240 T + 201957 T^{2} - 52171296 T^{3} + 201957 p^{3} T^{4} - 240 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 874 T + 389363 T^{2} - 137558140 T^{3} + 389363 p^{3} T^{4} - 874 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 948 T + 742897 T^{2} - 351969432 T^{3} + 742897 p^{3} T^{4} - 948 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 951 T + 777543 T^{2} - 412672466 T^{3} + 777543 p^{3} T^{4} - 951 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 742 T + 287377 T^{2} - 84916836 T^{3} + 287377 p^{3} T^{4} - 742 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 732 T + 1164141 T^{2} - 514616872 T^{3} + 1164141 p^{3} T^{4} - 732 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 585 T + 211531 T^{2} - 66466602 T^{3} + 211531 p^{3} T^{4} + 585 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 618 T + 1178605 T^{2} + 577905836 T^{3} + 1178605 p^{3} T^{4} + 618 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 774 T - 19175 T^{2} + 631907084 T^{3} - 19175 p^{3} T^{4} - 774 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 730 T + 1969183 T^{2} - 977258332 T^{3} + 1969183 p^{3} T^{4} - 730 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2518 T + 4687407 T^{2} - 5026558836 T^{3} + 4687407 p^{3} T^{4} - 2518 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.727606991301275538461913356755, −8.776572339520877250044150411411, −8.707232962088113423144034869413, −8.022987045468692264436945964334, −7.86259758057500287673712860966, −7.35749669280534342649152082388, −7.34014146228772072466719811718, −6.68155087205438743337154092528, −6.48403379467893319070742225867, −6.41058286286950942648420836216, −5.81237069493611635216232959761, −5.69511202612237312662261677269, −5.60811099853644938974540731664, −5.10262002890042975946948494123, −4.83034962277324929651688602657, −4.72368020022545884117791332098, −3.75886214893734267646206258598, −3.74126971840602204429961587063, −3.64249027771704366360354806105, −2.74675040156860044874362254419, −2.46267946334579441020744160968, −2.38667019151206458490095094702, −1.15821639174823189992608327213, −1.01953215186912515275215253221, −0.63470299444771382114383411091,
0.63470299444771382114383411091, 1.01953215186912515275215253221, 1.15821639174823189992608327213, 2.38667019151206458490095094702, 2.46267946334579441020744160968, 2.74675040156860044874362254419, 3.64249027771704366360354806105, 3.74126971840602204429961587063, 3.75886214893734267646206258598, 4.72368020022545884117791332098, 4.83034962277324929651688602657, 5.10262002890042975946948494123, 5.60811099853644938974540731664, 5.69511202612237312662261677269, 5.81237069493611635216232959761, 6.41058286286950942648420836216, 6.48403379467893319070742225867, 6.68155087205438743337154092528, 7.34014146228772072466719811718, 7.35749669280534342649152082388, 7.86259758057500287673712860966, 8.022987045468692264436945964334, 8.707232962088113423144034869413, 8.776572339520877250044150411411, 9.727606991301275538461913356755
