| L(s) = 1 | − 4·3-s − 8·7-s + 10·9-s + 8·13-s + 32·21-s − 16·23-s − 8·25-s − 20·27-s − 8·31-s + 8·37-s − 32·39-s − 16·47-s + 20·49-s + 16·59-s + 24·61-s − 80·63-s − 16·67-s + 64·69-s − 16·71-s − 8·73-s + 32·75-s − 24·79-s + 35·81-s − 8·89-s − 64·91-s + 32·93-s − 16·97-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.02·7-s + 10/3·9-s + 2.21·13-s + 6.98·21-s − 3.33·23-s − 8/5·25-s − 3.84·27-s − 1.43·31-s + 1.31·37-s − 5.12·39-s − 2.33·47-s + 20/7·49-s + 2.08·59-s + 3.07·61-s − 10.0·63-s − 1.95·67-s + 7.70·69-s − 1.89·71-s − 0.936·73-s + 3.69·75-s − 2.70·79-s + 35/9·81-s − 0.847·89-s − 6.70·91-s + 3.31·93-s − 1.62·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T^{2} + 16 T^{3} + 26 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T + 44 T^{2} + 24 p T^{3} + 510 T^{4} + 24 p^{2} T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 40 T^{2} - 136 T^{3} + 514 T^{4} - 136 p T^{5} + 40 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 36 T^{2} - 64 T^{3} + 614 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 44 T^{2} + 64 T^{3} + 918 T^{4} + 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 72 T^{2} + 112 T^{3} + 2426 T^{4} + 112 p T^{5} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T + 108 T^{2} + 616 T^{3} + 162 p T^{4} + 616 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 104 T^{2} - 776 T^{3} + 5346 T^{4} - 776 p T^{5} + 104 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 68 T^{2} + 64 T^{3} + 3206 T^{4} + 64 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 76 T^{2} - 64 T^{3} + 3830 T^{4} - 64 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 104 T^{2} + 272 T^{3} + 5594 T^{4} + 272 p T^{5} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 16 T + 204 T^{2} - 1552 T^{3} + 12758 T^{4} - 1552 p T^{5} + 204 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 24 T + 392 T^{2} - 4568 T^{3} + 40386 T^{4} - 4568 p T^{5} + 392 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 16 T + 172 T^{2} + 912 T^{3} + 6134 T^{4} + 912 p T^{5} + 172 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 16 T + 4 p T^{2} + 2640 T^{3} + 28070 T^{4} + 2640 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T + 172 T^{2} + 1720 T^{3} + 16006 T^{4} + 1720 p T^{5} + 172 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 24 T + 396 T^{2} + 4344 T^{3} + 42398 T^{4} + 4344 p T^{5} + 396 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 252 T^{2} + 128 T^{3} + 28598 T^{4} + 128 p T^{5} + 252 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T + 188 T^{2} + 2424 T^{3} + 17894 T^{4} + 2424 p T^{5} + 188 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 16 T + 356 T^{2} + 4400 T^{3} + 49990 T^{4} + 4400 p T^{5} + 356 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.54827942099288042734871500466, −6.32744752741606613608783378347, −6.15517108780854236076724719997, −5.92789306288816142461795522068, −5.91385954654644943504164181091, −5.61743476470865210520469075253, −5.53817591378049136181823929812, −5.29476200871291047121097128844, −5.27342616911954882065218689940, −4.58130487626457419467793132638, −4.48452698183479001738165444645, −4.35197401940095887374466607567, −4.04523083541068841246733150332, −3.74443548375038028090759043329, −3.63319785286207045399328138541, −3.61556688681850593422959095447, −3.60421932088888417508955849314, −2.98135627378011384545251074075, −2.64309888687020575128237689322, −2.40707773607869227519538103356, −2.38146626974936901309367501913, −1.58089658923940818515925108230, −1.52869885532175024578247111721, −1.34248449741149147163307503969, −1.02935264448947798211128901563, 0, 0, 0, 0,
1.02935264448947798211128901563, 1.34248449741149147163307503969, 1.52869885532175024578247111721, 1.58089658923940818515925108230, 2.38146626974936901309367501913, 2.40707773607869227519538103356, 2.64309888687020575128237689322, 2.98135627378011384545251074075, 3.60421932088888417508955849314, 3.61556688681850593422959095447, 3.63319785286207045399328138541, 3.74443548375038028090759043329, 4.04523083541068841246733150332, 4.35197401940095887374466607567, 4.48452698183479001738165444645, 4.58130487626457419467793132638, 5.27342616911954882065218689940, 5.29476200871291047121097128844, 5.53817591378049136181823929812, 5.61743476470865210520469075253, 5.91385954654644943504164181091, 5.92789306288816142461795522068, 6.15517108780854236076724719997, 6.32744752741606613608783378347, 6.54827942099288042734871500466
