| L(s) = 1 | − 4·5-s + 4·7-s + 8·13-s − 12·29-s + 12·31-s − 16·35-s + 16·37-s − 4·49-s − 20·53-s + 16·61-s − 32·65-s − 16·67-s + 8·71-s − 8·73-s + 12·79-s + 8·89-s + 32·91-s − 20·101-s + 4·103-s + 24·109-s + 8·113-s − 20·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s + 2.21·13-s − 2.22·29-s + 2.15·31-s − 2.70·35-s + 2.63·37-s − 4/7·49-s − 2.74·53-s + 2.04·61-s − 3.96·65-s − 1.95·67-s + 0.949·71-s − 0.936·73-s + 1.35·79-s + 0.847·89-s + 3.35·91-s − 1.99·101-s + 0.394·103-s + 2.29·109-s + 0.752·113-s − 1.81·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.094826700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.094826700\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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
−5.44522249754678150352922396842, −5.06290772074344357694882618383, −5.05614398351850211144988490634, −4.71402316773439943883224182030, −4.69994492696638575884327230411, −4.44979713557172732262389466134, −4.32629267628356351082820007230, −4.06253047098692020774931688828, −4.01133987193256310485214361431, −3.66393692288977074652037825721, −3.58976477539694952324328336651, −3.49070344139592879068885906983, −3.38713744081949611526101569657, −2.84222635600986726552435268695, −2.72491448978443404533273645287, −2.63453388733317289073244939599, −2.41471782207082450786164746016, −1.81603226532849095313832267296, −1.75534487576057498731978088792, −1.66370985886183010971994515903, −1.44184299645144067216379426220, −1.08646826038270750513462293632, −0.804413752885254026448738341983, −0.47144492850248835439305255937, −0.34810070370882689301693943648,
0.34810070370882689301693943648, 0.47144492850248835439305255937, 0.804413752885254026448738341983, 1.08646826038270750513462293632, 1.44184299645144067216379426220, 1.66370985886183010971994515903, 1.75534487576057498731978088792, 1.81603226532849095313832267296, 2.41471782207082450786164746016, 2.63453388733317289073244939599, 2.72491448978443404533273645287, 2.84222635600986726552435268695, 3.38713744081949611526101569657, 3.49070344139592879068885906983, 3.58976477539694952324328336651, 3.66393692288977074652037825721, 4.01133987193256310485214361431, 4.06253047098692020774931688828, 4.32629267628356351082820007230, 4.44979713557172732262389466134, 4.69994492696638575884327230411, 4.71402316773439943883224182030, 5.05614398351850211144988490634, 5.06290772074344357694882618383, 5.44522249754678150352922396842
