| L(s) = 1 | + 2·4-s + 5·7-s − 5·25-s + 10·28-s + 10·37-s − 5·43-s + 18·49-s − 8·64-s − 5·67-s + 4·79-s − 10·100-s + 19·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 20·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 10·172-s + 173-s − 25·175-s + ⋯ |
| L(s) = 1 | + 4-s + 1.88·7-s − 25-s + 1.88·28-s + 1.64·37-s − 0.762·43-s + 18/7·49-s − 64-s − 0.610·67-s + 0.450·79-s − 100-s + 1.81·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.64·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s − 0.762·172-s + 0.0760·173-s − 1.88·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.389720482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.389720482\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.666701525439978085811308809608, −8.941104923484224373050975615068, −8.546368934587619273226430073250, −7.968486753178926108508959829676, −7.56693266687474729542118165447, −7.30210105915125174102494200516, −6.48572745289681597519306363452, −6.06039738628443802450281714227, −5.45071950637189784126292696374, −4.80832105856080126763216830145, −4.37555080229996109256154673154, −3.61617877769699284494632874961, −2.59606963374854415313349396825, −2.07225487074358357683584536542, −1.31869238086390096442822435857,
1.31869238086390096442822435857, 2.07225487074358357683584536542, 2.59606963374854415313349396825, 3.61617877769699284494632874961, 4.37555080229996109256154673154, 4.80832105856080126763216830145, 5.45071950637189784126292696374, 6.06039738628443802450281714227, 6.48572745289681597519306363452, 7.30210105915125174102494200516, 7.56693266687474729542118165447, 7.968486753178926108508959829676, 8.546368934587619273226430073250, 8.941104923484224373050975615068, 9.666701525439978085811308809608
