| L(s) = 1 | + 4·5-s − 9-s − 12·11-s − 8·19-s + 11·25-s + 16·31-s − 12·41-s − 4·45-s + 10·49-s − 48·55-s + 12·59-s + 12·61-s + 8·71-s − 16·79-s + 81-s − 28·89-s − 32·95-s + 12·99-s − 16·101-s + 4·109-s + 86·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 3.61·11-s − 1.83·19-s + 11/5·25-s + 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s − 6.47·55-s + 1.56·59-s + 1.53·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 2.96·89-s − 3.28·95-s + 1.20·99-s − 1.59·101-s + 0.383·109-s + 7.81·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648629722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648629722\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.16301143950663709372774734815, −10.10843074733092975834423502054, −9.677742935051225656092075022861, −8.824488093944440135428863578877, −8.455995937978732418884194336711, −8.277596149553374920068614923576, −7.968926470881624564798648519485, −7.18008565536287803926850182462, −6.72118002178376890734929170176, −6.44912600717094385911057205493, −5.68519514885472871063497048270, −5.45498881352701941400877597056, −5.28413821514658668311735752097, −4.62606190514689230156777348903, −4.18954719814143513552709798199, −2.91049022848173918235475093563, −2.80070665029808225874580405786, −2.35021042146029618209966128923, −1.85377158794925610007949732812, −0.55855792703317023892837593089,
0.55855792703317023892837593089, 1.85377158794925610007949732812, 2.35021042146029618209966128923, 2.80070665029808225874580405786, 2.91049022848173918235475093563, 4.18954719814143513552709798199, 4.62606190514689230156777348903, 5.28413821514658668311735752097, 5.45498881352701941400877597056, 5.68519514885472871063497048270, 6.44912600717094385911057205493, 6.72118002178376890734929170176, 7.18008565536287803926850182462, 7.968926470881624564798648519485, 8.277596149553374920068614923576, 8.455995937978732418884194336711, 8.824488093944440135428863578877, 9.677742935051225656092075022861, 10.10843074733092975834423502054, 10.16301143950663709372774734815
