| L(s) = 1 | − 2·5-s + 4·9-s − 12·13-s − 4·17-s + 3·25-s + 8·29-s − 4·37-s − 8·45-s − 4·49-s − 12·53-s − 4·61-s + 24·65-s + 28·73-s + 7·81-s + 8·85-s − 20·89-s + 4·97-s + 20·109-s + 12·113-s − 48·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 4/3·9-s − 3.32·13-s − 0.970·17-s + 3/5·25-s + 1.48·29-s − 0.657·37-s − 1.19·45-s − 4/7·49-s − 1.64·53-s − 0.512·61-s + 2.97·65-s + 3.27·73-s + 7/9·81-s + 0.867·85-s − 2.11·89-s + 0.406·97-s + 1.91·109-s + 1.12·113-s − 4.43·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-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
| \(L(1)\) |
\(\approx\) |
\(0.7728737872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7728737872\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.946057881478745889890870650354, −9.566160174959817216423668238033, −9.111665618222320980757272165362, −8.721249657251882844686648243933, −8.019184288594452912756046717291, −7.74471321963778819249250167928, −7.56368984657157567406817199118, −6.85587211520480509434703192772, −6.83035964707492582337739090398, −6.45706808082448915766441525949, −5.51806833757027764191767716019, −4.93670135674997377284350863438, −4.67096239245022807699755717172, −4.58073876691353591712906041194, −3.87397682792607475365165086426, −3.28516378942071903774413828489, −2.57085970041309012426694135381, −2.29024230452580220115307216661, −1.46548890812263135956192567274, −0.36729420283950843009548792518,
0.36729420283950843009548792518, 1.46548890812263135956192567274, 2.29024230452580220115307216661, 2.57085970041309012426694135381, 3.28516378942071903774413828489, 3.87397682792607475365165086426, 4.58073876691353591712906041194, 4.67096239245022807699755717172, 4.93670135674997377284350863438, 5.51806833757027764191767716019, 6.45706808082448915766441525949, 6.83035964707492582337739090398, 6.85587211520480509434703192772, 7.56368984657157567406817199118, 7.74471321963778819249250167928, 8.019184288594452912756046717291, 8.721249657251882844686648243933, 9.111665618222320980757272165362, 9.566160174959817216423668238033, 9.946057881478745889890870650354
