| L(s) = 1 | − 2·5-s − 9-s − 4·11-s − 8·19-s − 25-s − 4·29-s + 4·31-s − 20·41-s + 2·45-s + 14·49-s + 8·55-s − 20·59-s + 20·61-s − 20·79-s + 81-s − 12·89-s + 16·95-s + 4·99-s − 12·101-s + 20·109-s − 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s − 1.20·11-s − 1.83·19-s − 1/5·25-s − 0.742·29-s + 0.718·31-s − 3.12·41-s + 0.298·45-s + 2·49-s + 1.07·55-s − 2.60·59-s + 2.56·61-s − 2.25·79-s + 1/9·81-s − 1.27·89-s + 1.64·95-s + 0.402·99-s − 1.19·101-s + 1.91·109-s − 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.862653069328900294902203271510, −8.450007369571122748142489196570, −8.278826299493909442646128934791, −8.026118965324219905115704444601, −7.27655235269514207531850519138, −7.25565955360981304513100174162, −6.69528160587130971565562916773, −6.23721875831596843815286731299, −5.72463504373034276653714353171, −5.46422574694358174010560560398, −4.76620266825936465418058900244, −4.62217490875236822024850053426, −3.93296243177478523102646233862, −3.66755594390135529130877505856, −3.06765707790770807966495293631, −2.49032777584496753482590794625, −2.11054689928198615821237636807, −1.31440268025933649203576461663, 0, 0,
1.31440268025933649203576461663, 2.11054689928198615821237636807, 2.49032777584496753482590794625, 3.06765707790770807966495293631, 3.66755594390135529130877505856, 3.93296243177478523102646233862, 4.62217490875236822024850053426, 4.76620266825936465418058900244, 5.46422574694358174010560560398, 5.72463504373034276653714353171, 6.23721875831596843815286731299, 6.69528160587130971565562916773, 7.25565955360981304513100174162, 7.27655235269514207531850519138, 8.026118965324219905115704444601, 8.278826299493909442646128934791, 8.450007369571122748142489196570, 8.862653069328900294902203271510
