| L(s) = 1 | + 2-s + 4-s − 8·7-s + 8-s − 2·9-s − 8·14-s + 16-s − 2·17-s − 2·18-s − 10·25-s − 8·28-s − 8·31-s + 32-s − 2·34-s − 2·36-s + 12·41-s + 34·49-s − 10·50-s − 8·56-s − 8·62-s + 16·63-s + 64-s − 2·68-s − 2·72-s + 4·73-s + 16·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 3.02·7-s + 0.353·8-s − 2/3·9-s − 2.13·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s − 2·25-s − 1.51·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/3·36-s + 1.87·41-s + 34/7·49-s − 1.41·50-s − 1.06·56-s − 1.01·62-s + 2.01·63-s + 1/8·64-s − 0.242·68-s − 0.235·72-s + 0.468·73-s + 1.80·79-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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.866374203841653278096661486834, −9.628355001583189625988551593246, −9.286071988264369821451329617999, −8.657255224248218342106471033370, −7.76589645819579710746731573989, −7.25946473788827804639135121523, −6.65330731261113455462227374582, −6.03835056135927571761510178023, −5.99911855171913780844071238816, −5.19013856842488788481068619038, −3.90229547122613769308947697962, −3.78997975332835388697662085523, −2.95627382514208231650511213543, −2.34610746208050944602353534104, 0,
2.34610746208050944602353534104, 2.95627382514208231650511213543, 3.78997975332835388697662085523, 3.90229547122613769308947697962, 5.19013856842488788481068619038, 5.99911855171913780844071238816, 6.03835056135927571761510178023, 6.65330731261113455462227374582, 7.25946473788827804639135121523, 7.76589645819579710746731573989, 8.657255224248218342106471033370, 9.286071988264369821451329617999, 9.628355001583189625988551593246, 9.866374203841653278096661486834
