| L(s) = 1 | − 2·5-s + 10·13-s − 10·17-s − 25-s − 8·29-s − 10·37-s − 16·41-s + 10·53-s − 20·65-s − 10·73-s − 9·81-s + 20·85-s + 10·97-s − 12·109-s + 30·113-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2.77·13-s − 2.42·17-s − 1/5·25-s − 1.48·29-s − 1.64·37-s − 2.49·41-s + 1.37·53-s − 2.48·65-s − 1.17·73-s − 81-s + 2.16·85-s + 1.01·97-s − 1.14·109-s + 2.82·113-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028080073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028080073\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11438865071386317091421550684, −9.091899945531123201644425725516, −9.030887611588143986310667730503, −8.585604864707263456467493059742, −8.453517746850719473485622194063, −7.991150076970423622472806007073, −7.34160995165524630651982063857, −6.85879016846338345739621394763, −6.79680445208458566379433301339, −5.97869096312774114471418214610, −5.96700528524974412324000094128, −5.18109618127051032617823362923, −4.74173148678129836489681767748, −3.97487450572934773574594098011, −3.91624449322451952782696936069, −3.49310370129605724756387465616, −2.84378413786553694923164123873, −1.78797119986679657755660265937, −1.70153620039890492054305451472, −0.42186522704871624138557909427,
0.42186522704871624138557909427, 1.70153620039890492054305451472, 1.78797119986679657755660265937, 2.84378413786553694923164123873, 3.49310370129605724756387465616, 3.91624449322451952782696936069, 3.97487450572934773574594098011, 4.74173148678129836489681767748, 5.18109618127051032617823362923, 5.96700528524974412324000094128, 5.97869096312774114471418214610, 6.79680445208458566379433301339, 6.85879016846338345739621394763, 7.34160995165524630651982063857, 7.991150076970423622472806007073, 8.453517746850719473485622194063, 8.585604864707263456467493059742, 9.030887611588143986310667730503, 9.091899945531123201644425725516, 10.11438865071386317091421550684
