| L(s) = 1 | − 9-s − 8·11-s + 16·19-s − 12·29-s − 12·41-s + 14·49-s + 24·59-s − 28·61-s + 16·79-s + 81-s − 4·89-s + 8·99-s + 28·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 16·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.41·11-s + 3.67·19-s − 2.22·29-s − 1.87·41-s + 2·49-s + 3.12·59-s − 3.58·61-s + 1.80·79-s + 1/9·81-s − 0.423·89-s + 0.804·99-s + 2.78·101-s + 2.68·109-s + 2.36·121-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 − 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022128074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022128074\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.298402926786489461057683659540, −8.046147729904683778177302242738, −7.64917687962721049091706370098, −7.43899866838358488122643349535, −7.13383000718153448427779313315, −6.97354995899325591917896807175, −6.00664242620333576473850816129, −5.79188931656388954270529950187, −5.50806447072765436742697522391, −5.28018375630313113262555174930, −4.85558668906089427779174144159, −4.62285788542302099364563090716, −3.63491282060755656464660176877, −3.56656064745155220738497737710, −2.97145665927498414875462796101, −2.91460347195458961443171633410, −1.97594621130449018447264489078, −1.95852100140149822520479181858, −0.897353534312629554796886580254, −0.47330227889760011102679101311,
0.47330227889760011102679101311, 0.897353534312629554796886580254, 1.95852100140149822520479181858, 1.97594621130449018447264489078, 2.91460347195458961443171633410, 2.97145665927498414875462796101, 3.56656064745155220738497737710, 3.63491282060755656464660176877, 4.62285788542302099364563090716, 4.85558668906089427779174144159, 5.28018375630313113262555174930, 5.50806447072765436742697522391, 5.79188931656388954270529950187, 6.00664242620333576473850816129, 6.97354995899325591917896807175, 7.13383000718153448427779313315, 7.43899866838358488122643349535, 7.64917687962721049091706370098, 8.046147729904683778177302242738, 8.298402926786489461057683659540
