| L(s) = 1 | − 2·5-s − 17·9-s + 96·13-s − 2·17-s + 51·25-s + 96·29-s + 98·37-s − 192·41-s + 34·45-s − 98·49-s + 50·53-s + 2·61-s − 192·65-s − 190·73-s + 81·81-s + 4·85-s − 190·89-s + 576·97-s − 146·101-s + 142·109-s + 384·113-s − 1.63e3·117-s + 47·121-s − 198·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2/5·5-s − 1.88·9-s + 7.38·13-s − 0.117·17-s + 2.03·25-s + 3.31·29-s + 2.64·37-s − 4.68·41-s + 0.755·45-s − 2·49-s + 0.943·53-s + 2/61·61-s − 2.95·65-s − 2.60·73-s + 81-s + 4/85·85-s − 2.13·89-s + 5.93·97-s − 1.44·101-s + 1.30·109-s + 3.39·113-s − 13.9·117-s + 0.388·121-s − 1.58·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.658145903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.658145903\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 17 T^{2} + 208 T^{4} + 17 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + T - 24 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 47 T^{2} - 12432 T^{4} - 47 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T - 288 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 673 T^{2} + 322608 T^{4} + 673 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 1009 T^{2} + 738240 T^{4} + 1009 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 241 T^{2} - 865440 T^{4} + 241 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 49 T + 1032 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1393 T^{2} - 2939232 T^{4} + 1393 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T - 2184 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 6673 T^{2} + 32411568 T^{4} + 6673 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T - 3720 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4753 T^{2} + 2439888 T^{4} + 4753 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 95 T + 3696 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 10801 T^{2} + 77711520 T^{4} + 10801 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 95 T + 1104 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{4} \) |
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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
−8.631523588927382222490841021281, −8.480687416624560540349841719584, −8.325856362904633612291798460258, −8.124221418843820805021395188875, −8.106572186596981669582269284407, −7.39233571834020532379012775333, −6.76453389539697833731534766449, −6.67258458886587069052492617636, −6.52037659626254115318793603819, −6.21939568193788991521427275247, −5.93200049368150219763611210724, −5.92970404734542269380602983807, −5.61482349692985381011255685271, −5.04927219900615403816340806687, −4.62780547793922233015425645479, −4.39927744703282405432001409888, −4.07116894065548118611616255955, −3.36540246700315755760902864489, −3.26480426475421587837184058987, −3.26141490713238760708481507861, −3.08224049187089788702005611500, −2.03452409131156817553221692859, −1.33638206571630237313296692738, −1.12026154097295026105710592308, −0.77767828322521165616781488047,
0.77767828322521165616781488047, 1.12026154097295026105710592308, 1.33638206571630237313296692738, 2.03452409131156817553221692859, 3.08224049187089788702005611500, 3.26141490713238760708481507861, 3.26480426475421587837184058987, 3.36540246700315755760902864489, 4.07116894065548118611616255955, 4.39927744703282405432001409888, 4.62780547793922233015425645479, 5.04927219900615403816340806687, 5.61482349692985381011255685271, 5.92970404734542269380602983807, 5.93200049368150219763611210724, 6.21939568193788991521427275247, 6.52037659626254115318793603819, 6.67258458886587069052492617636, 6.76453389539697833731534766449, 7.39233571834020532379012775333, 8.106572186596981669582269284407, 8.124221418843820805021395188875, 8.325856362904633612291798460258, 8.480687416624560540349841719584, 8.631523588927382222490841021281
