L(s) = 1 | + 2·9-s − 4·11-s + 12·19-s − 20·29-s + 16·31-s − 12·41-s + 14·49-s − 12·59-s − 12·61-s + 8·71-s + 16·79-s − 5·81-s − 28·89-s − 8·99-s − 12·101-s − 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
L(s) = 1 | + 2/3·9-s − 1.20·11-s + 2.75·19-s − 3.71·29-s + 2.87·31-s − 1.87·41-s + 2·49-s − 1.56·59-s − 1.53·61-s + 0.949·71-s + 1.80·79-s − 5/9·81-s − 2.96·89-s − 0.804·99-s − 1.19·101-s − 0.383·109-s − 0.909·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.843612674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.843612674\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.144159693947736191286244516298, −8.227816984977509911433058229045, −8.097018316602670618821515731506, −7.69248789678896525254147334679, −7.41702609876303403863479142148, −7.12368729949137090451396955280, −6.70492233694914886503415239654, −6.19769260995557303738731588797, −5.63342485915617040970445519948, −5.37035622879085587549859090777, −5.24305098578229765011167938827, −4.60681732883161380728293910812, −4.24237702485883785399979067475, −3.59972110320655939542041910319, −3.37682411199879473940620979403, −2.74745469251524297808161533447, −2.48834277529290382597762597028, −1.56223433044391204634848696590, −1.36106394902065450188981174780, −0.42962066641595907378571198064,
0.42962066641595907378571198064, 1.36106394902065450188981174780, 1.56223433044391204634848696590, 2.48834277529290382597762597028, 2.74745469251524297808161533447, 3.37682411199879473940620979403, 3.59972110320655939542041910319, 4.24237702485883785399979067475, 4.60681732883161380728293910812, 5.24305098578229765011167938827, 5.37035622879085587549859090777, 5.63342485915617040970445519948, 6.19769260995557303738731588797, 6.70492233694914886503415239654, 7.12368729949137090451396955280, 7.41702609876303403863479142148, 7.69248789678896525254147334679, 8.097018316602670618821515731506, 8.227816984977509911433058229045, 9.144159693947736191286244516298