| L(s) = 1 | − 2-s + 4-s − 8-s − 2·9-s − 8·13-s + 16-s + 2·17-s + 2·18-s − 4·19-s − 6·25-s + 8·26-s − 32-s − 2·34-s − 2·36-s + 4·38-s − 4·43-s − 2·49-s + 6·50-s − 8·52-s + 8·53-s − 20·59-s + 64-s − 4·67-s + 2·68-s + 2·72-s − 4·76-s − 5·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.917·19-s − 6/5·25-s + 1.56·26-s − 0.176·32-s − 0.342·34-s − 1/3·36-s + 0.648·38-s − 0.609·43-s − 2/7·49-s + 0.848·50-s − 1.10·52-s + 1.09·53-s − 2.60·59-s + 1/8·64-s − 0.488·67-s + 0.242·68-s + 0.235·72-s − 0.458·76-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.04816201898213158804411391639, −9.677729324561388756920590300289, −9.031999547244992859229019665711, −8.567826405181456282159198401306, −7.977291922485477404325036573972, −7.36201326366773707229211009757, −7.20889035915696995877004824232, −6.17650433034386681089394135505, −5.88063887989034217588631771396, −4.98874214236023196544129758180, −4.50421027727542554840801729526, −3.43331410730790939911193473980, −2.63736500428663363668808453369, −1.92635709516537142164401444085, 0,
1.92635709516537142164401444085, 2.63736500428663363668808453369, 3.43331410730790939911193473980, 4.50421027727542554840801729526, 4.98874214236023196544129758180, 5.88063887989034217588631771396, 6.17650433034386681089394135505, 7.20889035915696995877004824232, 7.36201326366773707229211009757, 7.977291922485477404325036573972, 8.567826405181456282159198401306, 9.031999547244992859229019665711, 9.677729324561388756920590300289, 10.04816201898213158804411391639
