| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 2·9-s − 2·14-s + 16-s − 2·18-s − 4·19-s − 10·25-s − 2·28-s + 32-s − 2·36-s + 4·37-s − 4·38-s − 16·43-s + 3·49-s − 10·50-s + 12·53-s − 2·56-s + 4·63-s + 64-s − 2·72-s + 4·74-s − 4·76-s − 16·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s − 0.471·18-s − 0.917·19-s − 2·25-s − 0.377·28-s + 0.176·32-s − 1/3·36-s + 0.657·37-s − 0.648·38-s − 2.43·43-s + 3/7·49-s − 1.41·50-s + 1.64·53-s − 0.267·56-s + 0.503·63-s + 1/8·64-s − 0.235·72-s + 0.464·74-s − 0.458·76-s − 1.80·79-s − 5/9·81-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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.888226006934558931566258496787, −8.182643116839158569055135937831, −8.145906800879012290031802042193, −7.29668061464756137953129595214, −6.82397964191468499058519060303, −6.42455331216463290634831658204, −5.76321350617402115136418955935, −5.62168100149868099368912779374, −4.85990751466979898816213085515, −4.09052365620160024762211243831, −3.82418745638366191622617161116, −3.03039761753313177141617611476, −2.48514181668809031419660323366, −1.67120553980965395706477292535, 0,
1.67120553980965395706477292535, 2.48514181668809031419660323366, 3.03039761753313177141617611476, 3.82418745638366191622617161116, 4.09052365620160024762211243831, 4.85990751466979898816213085515, 5.62168100149868099368912779374, 5.76321350617402115136418955935, 6.42455331216463290634831658204, 6.82397964191468499058519060303, 7.29668061464756137953129595214, 8.145906800879012290031802042193, 8.182643116839158569055135937831, 8.888226006934558931566258496787
