| L(s) = 1 | + 8·7-s − 8·13-s − 16·23-s − 8·25-s + 8·31-s − 8·37-s − 16·47-s + 20·49-s − 16·59-s − 24·61-s − 16·67-s − 16·71-s − 8·73-s + 24·79-s + 8·89-s − 64·91-s − 16·97-s − 16·101-s + 24·103-s − 16·107-s − 24·109-s + 8·113-s − 28·121-s − 16·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 2.21·13-s − 3.33·23-s − 8/5·25-s + 1.43·31-s − 1.31·37-s − 2.33·47-s + 20/7·49-s − 2.08·59-s − 3.07·61-s − 1.95·67-s − 1.89·71-s − 0.936·73-s + 2.70·79-s + 0.847·89-s − 6.70·91-s − 1.62·97-s − 1.59·101-s + 2.36·103-s − 1.54·107-s − 2.29·109-s + 0.752·113-s − 2.54·121-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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
−5.86350100999126382601908699032, −5.53872470122145378045055269994, −5.24564869021528211073373915546, −5.17929021222829505231545324668, −4.92753807539620679524477719051, −4.78028828485817050373696020885, −4.73546622840489348481698698930, −4.66872830113867590151973803393, −4.41005038419679448614121496755, −4.13557026154887687313426407861, −3.98339193367130843221693356100, −3.96867063721958662060878579118, −3.61888793136333424851096845837, −3.20972975765980126726721990679, −3.09561240874680195913430193185, −2.95494517854485938188432057147, −2.80296239696175813493460893133, −2.23073696435743740634222633343, −2.10084358229669943227593659841, −2.08783694473472480641157552017, −2.05054507156803413785585931403, −1.49887224021910274400914658027, −1.39770654515595401705486031378, −1.38354991528801901104336646350, −1.10320239994262060691076389474, 0, 0, 0, 0,
1.10320239994262060691076389474, 1.38354991528801901104336646350, 1.39770654515595401705486031378, 1.49887224021910274400914658027, 2.05054507156803413785585931403, 2.08783694473472480641157552017, 2.10084358229669943227593659841, 2.23073696435743740634222633343, 2.80296239696175813493460893133, 2.95494517854485938188432057147, 3.09561240874680195913430193185, 3.20972975765980126726721990679, 3.61888793136333424851096845837, 3.96867063721958662060878579118, 3.98339193367130843221693356100, 4.13557026154887687313426407861, 4.41005038419679448614121496755, 4.66872830113867590151973803393, 4.73546622840489348481698698930, 4.78028828485817050373696020885, 4.92753807539620679524477719051, 5.17929021222829505231545324668, 5.24564869021528211073373915546, 5.53872470122145378045055269994, 5.86350100999126382601908699032
