| L(s) = 1 | − 2-s + 4-s − 8-s − 2·13-s + 16-s − 25-s + 2·26-s − 32-s − 2·37-s − 24·47-s + 2·49-s + 50-s − 2·52-s − 2·61-s + 64-s − 24·71-s + 22·73-s + 2·74-s − 24·83-s + 24·94-s + 4·97-s − 2·98-s − 100-s + 2·104-s + 24·107-s + 22·109-s − 22·121-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.554·13-s + 1/4·16-s − 1/5·25-s + 0.392·26-s − 0.176·32-s − 0.328·37-s − 3.50·47-s + 2/7·49-s + 0.141·50-s − 0.277·52-s − 0.256·61-s + 1/8·64-s − 2.84·71-s + 2.57·73-s + 0.232·74-s − 2.63·83-s + 2.47·94-s + 0.406·97-s − 0.202·98-s − 0.0999·100-s + 0.196·104-s + 2.32·107-s + 2.10·109-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.878539655843769294773456314485, −8.278948375699901432058512087738, −8.033910030135530589399167933473, −7.29957842837664580142557578027, −7.16277862972928710155421440227, −6.29261335708839240986736404399, −6.20376328566111560075210542997, −5.29484236286034362173688893149, −4.91673812495623205848110187798, −4.25638665319375785757127060430, −3.45423455586722913101065597862, −2.94651413139897301734292877776, −2.12069515781404083792870585763, −1.39451507835808804252991831473, 0,
1.39451507835808804252991831473, 2.12069515781404083792870585763, 2.94651413139897301734292877776, 3.45423455586722913101065597862, 4.25638665319375785757127060430, 4.91673812495623205848110187798, 5.29484236286034362173688893149, 6.20376328566111560075210542997, 6.29261335708839240986736404399, 7.16277862972928710155421440227, 7.29957842837664580142557578027, 8.033910030135530589399167933473, 8.278948375699901432058512087738, 8.878539655843769294773456314485
