| L(s) = 1 | + 2-s + 4-s + 6·5-s + 8-s + 6·10-s − 2·13-s + 16-s + 6·17-s + 6·20-s + 17·25-s − 2·26-s − 18·29-s + 32-s + 6·34-s − 2·37-s + 6·40-s − 12·41-s + 2·49-s + 17·50-s − 2·52-s + 12·53-s − 18·58-s − 2·61-s + 64-s − 12·65-s + 6·68-s + 22·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 2.68·5-s + 0.353·8-s + 1.89·10-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 1.34·20-s + 17/5·25-s − 0.392·26-s − 3.34·29-s + 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.948·40-s − 1.87·41-s + 2/7·49-s + 2.40·50-s − 0.277·52-s + 1.64·53-s − 2.36·58-s − 0.256·61-s + 1/8·64-s − 1.48·65-s + 0.727·68-s + 2.57·73-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
| \(L(1)\) |
\(\approx\) |
\(4.407444505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.407444505\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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
−9.101601246278505892226541271616, −8.878539655843769294773456314485, −8.033910030135530589399167933473, −7.29957842837664580142557578027, −7.24345462277249174674167705434, −6.20376328566111560075210542997, −6.18735538198950166736525445007, −5.49090131952205180218546644094, −5.29484236286034362173688893149, −4.91673812495623205848110187798, −3.69589098584017718744945020548, −3.45423455586722913101065597862, −2.30016225881827778465495235815, −2.12069515781404083792870585763, −1.39451507835808804252991831473,
1.39451507835808804252991831473, 2.12069515781404083792870585763, 2.30016225881827778465495235815, 3.45423455586722913101065597862, 3.69589098584017718744945020548, 4.91673812495623205848110187798, 5.29484236286034362173688893149, 5.49090131952205180218546644094, 6.18735538198950166736525445007, 6.20376328566111560075210542997, 7.24345462277249174674167705434, 7.29957842837664580142557578027, 8.033910030135530589399167933473, 8.878539655843769294773456314485, 9.101601246278505892226541271616
