| L(s) = 1 | − 100·5-s − 124·9-s + 584·13-s − 760·17-s + 6.25e3·25-s + 168·29-s − 1.97e4·37-s + 4.76e4·41-s + 1.24e4·45-s + 2.70e4·49-s − 1.74e4·53-s + 6.20e4·61-s − 5.84e4·65-s + 1.19e5·73-s − 1.03e4·81-s + 7.60e4·85-s + 2.09e5·89-s − 5.91e4·97-s + 4.71e5·101-s + 1.82e5·109-s − 4.91e5·113-s − 7.24e4·117-s − 3.82e5·121-s − 3.12e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.510·9-s + 0.958·13-s − 0.637·17-s + 2·25-s + 0.0370·29-s − 2.37·37-s + 4.42·41-s + 0.912·45-s + 1.61·49-s − 0.855·53-s + 2.13·61-s − 1.71·65-s + 2.62·73-s − 0.174·81-s + 1.14·85-s + 2.80·89-s − 0.638·97-s + 4.59·101-s + 1.47·109-s − 3.62·113-s − 0.489·117-s − 2.37·121-s − 1.78·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.05051352602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05051352602\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 124 T^{2} + 2854 p^{2} T^{4} + 124 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 27060 T^{2} + 497649542 T^{4} - 27060 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 382252 T^{2} + 87516901782 T^{4} + 382252 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 292 T - 102402 T^{2} - 292 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 380 T + 2009510 T^{2} + 380 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 1633036 T^{2} + 154661897526 T^{4} + 1633036 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 25651084 T^{2} + 247347299659206 T^{4} + 25651084 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 84 T + 9837118 T^{2} - 84 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 24283452 T^{2} + 1637830022018822 T^{4} + 24283452 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 9868 T + 159567054 T^{2} + 9868 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 23812 T + 331016342 T^{2} - 23812 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 385757980 T^{2} + 71059469286049782 T^{4} + 385757980 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 353765740 T^{2} + 66158288717340582 T^{4} + 353765740 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 8748 T + 833865262 T^{2} + 8748 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 488747308 T^{2} + 896846029819225302 T^{4} + 488747308 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 31012 T + 1250446302 T^{2} - 31012 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 3336863100 T^{2} + 6361754088145172822 T^{4} + 3336863100 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 6374750812 T^{2} + 16622748841310481702 T^{4} + 6374750812 p^{10} T^{6} + p^{20} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 59700 T + 5029368950 T^{2} - 59700 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 10884258108 T^{2} + 48452581383610561862 T^{4} + 10884258108 p^{10} T^{6} + p^{20} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 7906443964 T^{2} + 42876570344126662806 T^{4} + 7906443964 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 104660 T + 13126874198 T^{2} - 104660 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 29564 T + 14772618438 T^{2} + 29564 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−7.73766332927026555168499950553, −7.48097292743649657689981089782, −6.90219499185813880955199822112, −6.84504241848964535985812351872, −6.76639429220028862214627252887, −6.33879927715144399269539489634, −5.87561598130170726797184463807, −5.71908440615823920344665686573, −5.71152612535789906180713443383, −4.94364137461186506208857842252, −4.90269530033245984248830644931, −4.70603257975404462431430792545, −4.11123233772482731093033657140, −4.10644166766514941473674323483, −3.53139586430385769395510948998, −3.52251237775632288305633151349, −3.48502115920092523732939073655, −2.65860537927123401593317339909, −2.41324136384855208279012798790, −2.30787696412690187814038619590, −1.77691291437023557513774128679, −0.985418196536609668154811242791, −0.963874473709784401636316798397, −0.70083149909481472445654273680, −0.03394862889853045545819915795,
0.03394862889853045545819915795, 0.70083149909481472445654273680, 0.963874473709784401636316798397, 0.985418196536609668154811242791, 1.77691291437023557513774128679, 2.30787696412690187814038619590, 2.41324136384855208279012798790, 2.65860537927123401593317339909, 3.48502115920092523732939073655, 3.52251237775632288305633151349, 3.53139586430385769395510948998, 4.10644166766514941473674323483, 4.11123233772482731093033657140, 4.70603257975404462431430792545, 4.90269530033245984248830644931, 4.94364137461186506208857842252, 5.71152612535789906180713443383, 5.71908440615823920344665686573, 5.87561598130170726797184463807, 6.33879927715144399269539489634, 6.76639429220028862214627252887, 6.84504241848964535985812351872, 6.90219499185813880955199822112, 7.48097292743649657689981089782, 7.73766332927026555168499950553
