| L(s) = 1 | + 4·3-s + 4·5-s + 4·7-s + 10·9-s + 8·13-s + 16·15-s + 16·21-s + 20·27-s + 12·29-s + 12·31-s + 16·35-s + 16·37-s + 32·39-s + 40·45-s − 4·49-s + 20·53-s + 16·61-s + 40·63-s + 32·65-s − 16·67-s − 8·71-s − 8·73-s + 12·79-s + 35·81-s + 48·87-s − 8·89-s + 32·91-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s + 1.51·7-s + 10/3·9-s + 2.21·13-s + 4.13·15-s + 3.49·21-s + 3.84·27-s + 2.22·29-s + 2.15·31-s + 2.70·35-s + 2.63·37-s + 5.12·39-s + 5.96·45-s − 4/7·49-s + 2.74·53-s + 2.04·61-s + 5.03·63-s + 3.96·65-s − 1.95·67-s − 0.949·71-s − 0.936·73-s + 1.35·79-s + 35/9·81-s + 5.14·87-s − 0.847·89-s + 3.35·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(55.86299808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(55.86299808\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 16 T^{2} - 44 T^{3} + 118 T^{4} - 44 p T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 20 T^{2} - 60 T^{3} + 186 T^{4} - 60 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T^{2} + 32 T^{3} + 230 T^{4} + 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 56 T^{2} - 264 T^{3} + 1122 T^{4} - 264 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 44 T^{2} + 64 T^{3} + 966 T^{4} + 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 144 T^{2} - 964 T^{3} + 6422 T^{4} - 964 p T^{5} + 144 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 200 T^{2} - 1552 T^{3} + 11010 T^{4} - 1552 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 100 T^{2} + 192 T^{3} + 4726 T^{4} + 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 76 T^{2} + 256 T^{3} + 2726 T^{4} + 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 336 T^{2} - 3452 T^{3} + 30134 T^{4} - 3452 p T^{5} + 336 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 296 T^{2} - 2704 T^{3} + 27618 T^{4} - 2704 p T^{5} + 296 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 4 p T^{2} + 2960 T^{3} + 27190 T^{4} + 2960 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 252 T^{2} + 1512 T^{3} + 25766 T^{4} + 1512 p T^{5} + 252 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 116 T^{2} + 160 T^{3} + 5510 T^{4} + 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 156 T^{2} + 504 T^{3} + 10022 T^{4} + 504 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.30504788559228919638741647124, −5.91536354499007353720222245600, −5.71990044309529154143982728813, −5.68954389710944627044267913818, −5.53980266290598741579356042350, −4.89871330476596583375859166434, −4.88624194717432860137728190886, −4.64994958964762952258666766866, −4.61583065950290170096756665315, −4.17520230710405257669018800579, −4.04937066593116514048240905307, −3.93399625159702099212922007498, −3.75615190083247744066794255490, −3.09907401221153140040081363078, −3.05648103892727925533859993977, −3.05089411415083314282680706107, −2.80134687702366256094082924920, −2.25143499938521634391556774143, −2.15814823050182196297447771147, −2.02424011702061067180305134123, −1.94999727565317841272604993228, −1.31055387821963451636965326312, −1.15424370902459972779351263586, −1.03787889492982455688881607384, −0.78148081609838184363324004548,
0.78148081609838184363324004548, 1.03787889492982455688881607384, 1.15424370902459972779351263586, 1.31055387821963451636965326312, 1.94999727565317841272604993228, 2.02424011702061067180305134123, 2.15814823050182196297447771147, 2.25143499938521634391556774143, 2.80134687702366256094082924920, 3.05089411415083314282680706107, 3.05648103892727925533859993977, 3.09907401221153140040081363078, 3.75615190083247744066794255490, 3.93399625159702099212922007498, 4.04937066593116514048240905307, 4.17520230710405257669018800579, 4.61583065950290170096756665315, 4.64994958964762952258666766866, 4.88624194717432860137728190886, 4.89871330476596583375859166434, 5.53980266290598741579356042350, 5.68954389710944627044267913818, 5.71990044309529154143982728813, 5.91536354499007353720222245600, 6.30504788559228919638741647124
