| L(s) = 1 | − 2·3-s + 4·5-s − 2·7-s + 2·9-s − 6·13-s − 8·15-s − 6·17-s − 12·19-s + 4·21-s − 6·23-s + 11·25-s − 6·27-s − 8·35-s − 6·37-s + 12·39-s − 12·41-s − 6·43-s + 8·45-s + 18·47-s + 2·49-s + 12·51-s − 10·53-s + 24·57-s − 20·59-s + 24·61-s − 4·63-s − 24·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.755·7-s + 2/3·9-s − 1.66·13-s − 2.06·15-s − 1.45·17-s − 2.75·19-s + 0.872·21-s − 1.25·23-s + 11/5·25-s − 1.15·27-s − 1.35·35-s − 0.986·37-s + 1.92·39-s − 1.87·41-s − 0.914·43-s + 1.19·45-s + 2.62·47-s + 2/7·49-s + 1.68·51-s − 1.37·53-s + 3.17·57-s − 2.60·59-s + 3.07·61-s − 0.503·63-s − 2.97·65-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.606299836933068250284171331592, −9.199410844352709356249726533898, −8.632318355985944144310071390822, −8.573553673398465119919932228310, −7.74162940427514495117161920140, −7.14168588225264393809283129143, −6.74614447926115849110520562244, −6.55790017168689678450898063968, −5.98330834278771211460823296822, −5.97872457856260773200563781721, −5.24034670808618061461712700234, −4.99151588020140837812434783222, −4.23933687504990012576776971278, −4.20491201335763035945327198834, −3.10793775980364593750376803825, −2.46298842795261715058988739993, −1.98380104940191069438811831423, −1.76459039528730533636464057857, 0, 0,
1.76459039528730533636464057857, 1.98380104940191069438811831423, 2.46298842795261715058988739993, 3.10793775980364593750376803825, 4.20491201335763035945327198834, 4.23933687504990012576776971278, 4.99151588020140837812434783222, 5.24034670808618061461712700234, 5.97872457856260773200563781721, 5.98330834278771211460823296822, 6.55790017168689678450898063968, 6.74614447926115849110520562244, 7.14168588225264393809283129143, 7.74162940427514495117161920140, 8.573553673398465119919932228310, 8.632318355985944144310071390822, 9.199410844352709356249726533898, 9.606299836933068250284171331592
