| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·9-s + 6·11-s + 2·14-s + 16-s − 2·18-s − 6·22-s − 8·23-s − 6·25-s − 2·28-s + 2·29-s − 32-s + 2·36-s − 14·37-s − 24·43-s + 6·44-s + 8·46-s − 3·49-s + 6·50-s − 2·53-s + 2·56-s − 2·58-s − 4·63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 2/3·9-s + 1.80·11-s + 0.534·14-s + 1/4·16-s − 0.471·18-s − 1.27·22-s − 1.66·23-s − 6/5·25-s − 0.377·28-s + 0.371·29-s − 0.176·32-s + 1/3·36-s − 2.30·37-s − 3.65·43-s + 0.904·44-s + 1.17·46-s − 3/7·49-s + 0.848·50-s − 0.274·53-s + 0.267·56-s − 0.262·58-s − 0.503·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−9.018682906620481752014599990941, −8.349249841237217182629583731243, −8.143784062139351449482237870397, −7.40837288635255072786729733723, −6.84621102974705990857199295903, −6.59154190949514339166122867596, −6.20978114468454553535711113110, −5.56741659372590213451008161172, −4.81138687880925457081874221369, −4.09052251977104089331058220228, −3.61411922827203394131219212900, −3.16373906199664712104606801104, −1.80306662958575147659094810611, −1.64121605577949130548757500692, 0,
1.64121605577949130548757500692, 1.80306662958575147659094810611, 3.16373906199664712104606801104, 3.61411922827203394131219212900, 4.09052251977104089331058220228, 4.81138687880925457081874221369, 5.56741659372590213451008161172, 6.20978114468454553535711113110, 6.59154190949514339166122867596, 6.84621102974705990857199295903, 7.40837288635255072786729733723, 8.143784062139351449482237870397, 8.349249841237217182629583731243, 9.018682906620481752014599990941
