| L(s) = 1 | − 2·4-s − 2·7-s − 2·9-s + 4·16-s − 6·17-s − 25-s + 4·28-s − 8·31-s + 4·36-s − 12·41-s − 6·47-s − 11·49-s + 4·63-s − 8·64-s + 12·68-s + 12·71-s − 14·73-s + 16·79-s − 5·81-s + 24·89-s + 16·97-s + 2·100-s + 28·103-s − 8·112-s + 12·113-s + 12·119-s − 13·121-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 2/3·9-s + 16-s − 1.45·17-s − 1/5·25-s + 0.755·28-s − 1.43·31-s + 2/3·36-s − 1.87·41-s − 0.875·47-s − 1.57·49-s + 0.503·63-s − 64-s + 1.45·68-s + 1.42·71-s − 1.63·73-s + 1.80·79-s − 5/9·81-s + 2.54·89-s + 1.62·97-s + 1/5·100-s + 2.75·103-s − 0.755·112-s + 1.12·113-s + 1.10·119-s − 1.18·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.24320445923116880350231460909, −10.02752729817219727448938419961, −9.180168549070062598241568188033, −9.035139763634884351001508227661, −8.457332600420497477897703269253, −7.86885586734404145925295723068, −7.15811652768508964807251071197, −6.39084306102520193435942181609, −6.05813289271022268292942638231, −5.03912355415234199771433602492, −4.80419327140120764096716079767, −3.67310116806022314895429635564, −3.31935920587958999413340629739, −2.04259190544357015606449170426, 0,
2.04259190544357015606449170426, 3.31935920587958999413340629739, 3.67310116806022314895429635564, 4.80419327140120764096716079767, 5.03912355415234199771433602492, 6.05813289271022268292942638231, 6.39084306102520193435942181609, 7.15811652768508964807251071197, 7.86885586734404145925295723068, 8.457332600420497477897703269253, 9.035139763634884351001508227661, 9.180168549070062598241568188033, 10.02752729817219727448938419961, 10.24320445923116880350231460909
