| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s − 5·9-s + 2·10-s − 2·13-s + 16-s − 4·17-s − 5·18-s + 2·20-s − 7·25-s − 2·26-s − 20·29-s + 32-s − 4·34-s − 5·36-s − 4·37-s + 2·40-s + 4·41-s − 10·45-s − 5·49-s − 7·50-s − 2·52-s + 8·53-s − 20·58-s + 24·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 5/3·9-s + 0.632·10-s − 0.554·13-s + 1/4·16-s − 0.970·17-s − 1.17·18-s + 0.447·20-s − 7/5·25-s − 0.392·26-s − 3.71·29-s + 0.176·32-s − 0.685·34-s − 5/6·36-s − 0.657·37-s + 0.316·40-s + 0.624·41-s − 1.49·45-s − 5/7·49-s − 0.989·50-s − 0.277·52-s + 1.09·53-s − 2.62·58-s + 3.07·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−8.747141121901939638246702173369, −8.623992312917426485212718662788, −7.64863866033487331303243767551, −7.50453196645377754362608030446, −6.84410476101872509067957385637, −6.11586748167354289065343430906, −5.89085517297907838217443853632, −5.34006059048988750874758197539, −5.19332401510978287808819247709, −4.08090613362146692342747948187, −3.77244210214899187409248707106, −2.97225582177936414794423792629, −2.07407948897858183398493397476, −2.06983106012671431074676520675, 0,
2.06983106012671431074676520675, 2.07407948897858183398493397476, 2.97225582177936414794423792629, 3.77244210214899187409248707106, 4.08090613362146692342747948187, 5.19332401510978287808819247709, 5.34006059048988750874758197539, 5.89085517297907838217443853632, 6.11586748167354289065343430906, 6.84410476101872509067957385637, 7.50453196645377754362608030446, 7.64863866033487331303243767551, 8.623992312917426485212718662788, 8.747141121901939638246702173369
