| L(s) = 1 | + 2·5-s + 10·13-s − 6·17-s − 25-s + 10·37-s + 16·41-s + 10·53-s + 20·61-s + 20·65-s − 22·73-s − 9·81-s − 12·85-s − 26·97-s + 40·101-s − 2·113-s + 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.77·13-s − 1.45·17-s − 1/5·25-s + 1.64·37-s + 2.49·41-s + 1.37·53-s + 2.56·61-s + 2.48·65-s − 2.57·73-s − 81-s − 1.30·85-s − 2.63·97-s + 3.98·101-s − 0.188·113-s + 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-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
| \(L(1)\) |
\(\approx\) |
\(3.251074648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.251074648\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.723196625988769977073010707511, −9.572547343624199286594981829627, −8.836477194116578605769590903401, −8.793143749449504087225407926829, −8.438532084563974586383765219105, −7.965993074617059378273552097239, −7.28375245413445627748022716670, −7.04797218311230918321979412655, −6.33117992061355912321257652891, −6.12828574381362666495716000787, −5.79013085366248113449353587587, −5.55199260923311550365024425552, −4.58817148866811050811047689168, −4.28484604778766262744419441521, −3.87166037316800843598702567159, −3.32881179075263808926235828517, −2.50565227736795968017560392463, −2.22865221693163064379198005795, −1.36960577918542225204050749125, −0.843167161494615464574820590539,
0.843167161494615464574820590539, 1.36960577918542225204050749125, 2.22865221693163064379198005795, 2.50565227736795968017560392463, 3.32881179075263808926235828517, 3.87166037316800843598702567159, 4.28484604778766262744419441521, 4.58817148866811050811047689168, 5.55199260923311550365024425552, 5.79013085366248113449353587587, 6.12828574381362666495716000787, 6.33117992061355912321257652891, 7.04797218311230918321979412655, 7.28375245413445627748022716670, 7.965993074617059378273552097239, 8.438532084563974586383765219105, 8.793143749449504087225407926829, 8.836477194116578605769590903401, 9.572547343624199286594981829627, 9.723196625988769977073010707511
