| L(s) = 1 | − 640·11-s − 1.40e3·13-s + 1.45e4·23-s − 1.51e5·25-s − 1.31e4·37-s + 7.09e5·47-s − 1.64e6·49-s + 3.06e6·59-s − 2.64e5·61-s + 7.98e6·71-s + 1.63e6·73-s + 1.84e7·83-s + 6.36e6·97-s + 6.74e7·107-s − 1.19e7·109-s − 2.56e7·121-s + ⋯ |
| L(s) = 1 | − 0.144·11-s − 0.176·13-s + 0.250·23-s − 1.94·25-s − 0.0427·37-s + 0.997·47-s − 2.00·49-s + 1.94·59-s − 0.149·61-s + 2.64·71-s + 0.491·73-s + 3.53·83-s + 0.708·97-s + 5.32·107-s − 0.886·109-s − 1.31·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.671155762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.671155762\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2 \wr C_2$ | \( 1 + 151572 T^{2} + 568112294 p^{2} T^{4} + 151572 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 1648988 T^{2} + 1526127215334 T^{4} + 1648988 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 320 T + 12973958 T^{2} + 320 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 700 T + 1655046 p T^{2} + 700 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 53528764 T^{2} + 94005192820378182 T^{4} - 53528764 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 627281900 T^{2} - 415796730785898858 T^{4} + 627281900 p^{14} T^{6} + p^{28} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 7296 T + 4220360398 T^{2} - 7296 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 53499213492 T^{2} + \)\(12\!\cdots\!78\)\( T^{4} + 53499213492 p^{14} T^{6} + p^{28} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 25861402372 T^{2} + \)\(16\!\cdots\!38\)\( T^{4} - 25861402372 p^{14} T^{6} + p^{28} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6580 T + 189770474430 T^{2} + 6580 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 210306458340 T^{2} + \)\(53\!\cdots\!22\)\( T^{4} + 210306458340 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 564493630604 T^{2} + \)\(22\!\cdots\!02\)\( T^{4} + 564493630604 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 7552 p T + 1021319166110 T^{2} - 7552 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 1048237870932 T^{2} + \)\(39\!\cdots\!94\)\( T^{4} + 1048237870932 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 1531520 T + 3126722309414 T^{2} - 1531520 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 132132 T + 5537699450798 T^{2} + 132132 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 11408315918636 T^{2} + \)\(78\!\cdots\!82\)\( T^{4} + 11408315918636 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3991040 T + 19372360924526 T^{2} - 3991040 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 816300 T + 12256995211094 T^{2} - 816300 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 38820993405500 T^{2} + \)\(93\!\cdots\!62\)\( T^{4} + 38820993405500 p^{14} T^{6} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 9213504 T + 51757917911158 T^{2} - 9213504 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 145151685502116 T^{2} + \)\(91\!\cdots\!46\)\( T^{4} + 145151685502116 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 3184220 T + 130070241012102 T^{2} - 3184220 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−7.57193112552794305370159231697, −6.98032990132853136098652680729, −6.65746496624789690228154096836, −6.53962963549234105497198252575, −6.51844191324728397728307990235, −5.93607876723418184675001126458, −5.66534899232507391001663892984, −5.47930964993229931197061834517, −5.36474202293231867347824249324, −4.85325965912171482062623244435, −4.52876632020684835575066010476, −4.51651072076636021758893599617, −4.07829285083067179418778308850, −3.62197141725277279520238332390, −3.39590007598657046714195408537, −3.37019023730072251958377167938, −2.93684961941275754171699221200, −2.24938555845579179837871862538, −2.14855298568658337714171321906, −2.05859639180041371949101091715, −1.72097672291728313464227497347, −1.10382399069703095402707420299, −0.74524145667103093231629323594, −0.52658483422704981550796752034, −0.36953462756951885911230776761,
0.36953462756951885911230776761, 0.52658483422704981550796752034, 0.74524145667103093231629323594, 1.10382399069703095402707420299, 1.72097672291728313464227497347, 2.05859639180041371949101091715, 2.14855298568658337714171321906, 2.24938555845579179837871862538, 2.93684961941275754171699221200, 3.37019023730072251958377167938, 3.39590007598657046714195408537, 3.62197141725277279520238332390, 4.07829285083067179418778308850, 4.51651072076636021758893599617, 4.52876632020684835575066010476, 4.85325965912171482062623244435, 5.36474202293231867347824249324, 5.47930964993229931197061834517, 5.66534899232507391001663892984, 5.93607876723418184675001126458, 6.51844191324728397728307990235, 6.53962963549234105497198252575, 6.65746496624789690228154096836, 6.98032990132853136098652680729, 7.57193112552794305370159231697
