| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·14-s + 16-s − 6·19-s + 8·25-s − 2·28-s + 2·29-s − 32-s + 6·38-s + 8·41-s − 16·43-s − 10·49-s − 8·50-s + 2·56-s − 2·58-s − 2·59-s + 6·61-s + 64-s + 6·71-s − 6·76-s − 8·82-s + 16·86-s + 10·89-s + 10·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.534·14-s + 1/4·16-s − 1.37·19-s + 8/5·25-s − 0.377·28-s + 0.371·29-s − 0.176·32-s + 0.973·38-s + 1.24·41-s − 2.43·43-s − 1.42·49-s − 1.13·50-s + 0.267·56-s − 0.262·58-s − 0.260·59-s + 0.768·61-s + 1/8·64-s + 0.712·71-s − 0.688·76-s − 0.883·82-s + 1.72·86-s + 1.05·89-s + 1.01·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
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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
−8.398823483190744146217651821647, −8.016384861275495379076985025064, −7.42899009308018658139916282541, −6.89045632455211484787069836670, −6.47440810931093094023095844439, −6.33930283748027030299687029081, −5.64779493322699939953167812111, −4.85100038302376077256902144274, −4.68694833798792249300304403875, −3.72714926485184399877035244487, −3.32429540076631449549502673738, −2.65404541298085988373493722320, −2.05275954718861231342390797751, −1.12289484532166136765229877300, 0,
1.12289484532166136765229877300, 2.05275954718861231342390797751, 2.65404541298085988373493722320, 3.32429540076631449549502673738, 3.72714926485184399877035244487, 4.68694833798792249300304403875, 4.85100038302376077256902144274, 5.64779493322699939953167812111, 6.33930283748027030299687029081, 6.47440810931093094023095844439, 6.89045632455211484787069836670, 7.42899009308018658139916282541, 8.016384861275495379076985025064, 8.398823483190744146217651821647
