| L(s) = 1 | − 3·2-s + 2·4-s − 12·5-s + 7-s + 3·8-s + 36·10-s + 24·11-s − 11·13-s − 3·14-s − 3·16-s + 70·19-s − 24·20-s − 72·22-s − 12·23-s + 71·25-s + 33·26-s + 2·28-s − 48·29-s + 25·31-s + 12·32-s − 12·35-s + 70·37-s − 210·38-s − 36·40-s + 132·41-s + 37·43-s + 48·44-s + ⋯ |
| L(s) = 1 | − 3/2·2-s + 1/2·4-s − 2.39·5-s + 1/7·7-s + 3/8·8-s + 18/5·10-s + 2.18·11-s − 0.846·13-s − 0.214·14-s − 0.187·16-s + 3.68·19-s − 6/5·20-s − 3.27·22-s − 0.521·23-s + 2.83·25-s + 1.26·26-s + 1/14·28-s − 1.65·29-s + 0.806·31-s + 3/8·32-s − 0.342·35-s + 1.89·37-s − 5.52·38-s − 0.899·40-s + 3.21·41-s + 0.860·43-s + 1.09·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6105783975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6105783975\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T - 48 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 24 T + 313 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 11 T - 48 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 146 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 577 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 48 T + 1609 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 25 T - 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 132 T + 7489 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 37 T - 480 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 96 T + 5281 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 3529 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 22 T - 4005 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 6937 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 11954 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 9408 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.96664004175185710409250429575, −11.62787100554256204354033096625, −11.20827583178880025422987017004, −10.86382525251767844181629585489, −9.777536133691671470174605492520, −9.584873284354326820765872875339, −9.138987007126477686293546488882, −8.995975232422786895614792494615, −8.036658528496187588092341653678, −7.68501642556987697615343101122, −7.41711928847179592547306793763, −7.27175148936554431104962408660, −6.11616858078856981127877152294, −5.54155822495402769592966055367, −4.39831633990062212528511327577, −4.19264682851245911260111482078, −3.60717958395139328154119408918, −2.79047376430411492569287317846, −0.930749589443212195966395082921, −0.815603160678585936346858678909,
0.815603160678585936346858678909, 0.930749589443212195966395082921, 2.79047376430411492569287317846, 3.60717958395139328154119408918, 4.19264682851245911260111482078, 4.39831633990062212528511327577, 5.54155822495402769592966055367, 6.11616858078856981127877152294, 7.27175148936554431104962408660, 7.41711928847179592547306793763, 7.68501642556987697615343101122, 8.036658528496187588092341653678, 8.995975232422786895614792494615, 9.138987007126477686293546488882, 9.584873284354326820765872875339, 9.777536133691671470174605492520, 10.86382525251767844181629585489, 11.20827583178880025422987017004, 11.62787100554256204354033096625, 11.96664004175185710409250429575
