| L(s) = 1 | + 6·9-s − 12·11-s + 12·19-s − 6·29-s − 6·31-s + 18·41-s + 13·49-s − 2·59-s + 16·61-s − 10·71-s + 27·81-s − 8·89-s − 72·99-s − 18·101-s + 12·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 72·171-s + ⋯ |
| L(s) = 1 | + 2·9-s − 3.61·11-s + 2.75·19-s − 1.11·29-s − 1.07·31-s + 2.81·41-s + 13/7·49-s − 0.260·59-s + 2.04·61-s − 1.18·71-s + 3·81-s − 0.847·89-s − 7.23·99-s − 1.79·101-s + 1.14·109-s + 7.81·121-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 + 1.69·169-s + 5.50·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.667481728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.667481728\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.332500620763886995163268329317, −7.974684030038371362661499655364, −7.50760568985551690456235254807, −7.46721978945700448306441662723, −7.32455680407539074097898174167, −7.05327383951325524737312396510, −6.23306749822150298562826978274, −5.70561057142759781889338646595, −5.42755694932790245043144064116, −5.35202882263727020787012978441, −4.90217461302165383951306066785, −4.43718168230733038584775330490, −3.96414004902763441188966460948, −3.63685219384694221354548819077, −2.91103777254490309407546115060, −2.75557830537893932966884538548, −2.23419283449759969299113854842, −1.73684536868741167577433724392, −0.992050436900914754116202054484, −0.52597335362696331861488673489,
0.52597335362696331861488673489, 0.992050436900914754116202054484, 1.73684536868741167577433724392, 2.23419283449759969299113854842, 2.75557830537893932966884538548, 2.91103777254490309407546115060, 3.63685219384694221354548819077, 3.96414004902763441188966460948, 4.43718168230733038584775330490, 4.90217461302165383951306066785, 5.35202882263727020787012978441, 5.42755694932790245043144064116, 5.70561057142759781889338646595, 6.23306749822150298562826978274, 7.05327383951325524737312396510, 7.32455680407539074097898174167, 7.46721978945700448306441662723, 7.50760568985551690456235254807, 7.974684030038371362661499655364, 8.332500620763886995163268329317
