| L(s) = 1 | − 2·5-s − 3·7-s − 4·16-s − 16·17-s + 3·25-s + 6·35-s + 10·37-s − 20·41-s + 8·43-s + 8·47-s + 2·49-s − 16·59-s − 18·67-s − 6·79-s + 8·80-s + 12·83-s + 32·85-s − 24·89-s − 20·109-s + 12·112-s + 48·119-s − 18·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s − 16-s − 3.88·17-s + 3/5·25-s + 1.01·35-s + 1.64·37-s − 3.12·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s − 2.08·59-s − 2.19·67-s − 0.675·79-s + 0.894·80-s + 1.31·83-s + 3.47·85-s − 2.54·89-s − 1.91·109-s + 1.13·112-s + 4.40·119-s − 1.63·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.44584033135863585609716971257, −7.43336263509929639505497494700, −6.68799725367635564271959497809, −6.47641760141699532227966754862, −6.31521897643073047118022949368, −5.48591205290013748944065824002, −4.69221690938344723805408149899, −4.46062104683117077283188560086, −4.14431032778878436283934576248, −3.51283869966402115574490748912, −2.65254993398592503147695516254, −2.55983377045085713330071090790, −1.61576636967617660372099312199, 0, 0,
1.61576636967617660372099312199, 2.55983377045085713330071090790, 2.65254993398592503147695516254, 3.51283869966402115574490748912, 4.14431032778878436283934576248, 4.46062104683117077283188560086, 4.69221690938344723805408149899, 5.48591205290013748944065824002, 6.31521897643073047118022949368, 6.47641760141699532227966754862, 6.68799725367635564271959497809, 7.43336263509929639505497494700, 7.44584033135863585609716971257
