| L(s) = 1 | − 2-s + 4-s − 8-s − 2·13-s + 16-s − 9·23-s + 8·25-s + 2·26-s − 32-s − 11·37-s + 9·46-s + 3·47-s + 2·49-s − 8·50-s − 2·52-s − 18·59-s + 7·61-s + 64-s + 3·71-s − 14·73-s + 11·74-s − 15·83-s − 9·92-s − 3·94-s + 13·97-s − 2·98-s + 8·100-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.554·13-s + 1/4·16-s − 1.87·23-s + 8/5·25-s + 0.392·26-s − 0.176·32-s − 1.80·37-s + 1.32·46-s + 0.437·47-s + 2/7·49-s − 1.13·50-s − 0.277·52-s − 2.34·59-s + 0.896·61-s + 1/8·64-s + 0.356·71-s − 1.63·73-s + 1.27·74-s − 1.64·83-s − 0.938·92-s − 0.309·94-s + 1.31·97-s − 0.202·98-s + 4/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.683059896151617221990799357979, −8.576014478069992620504614276097, −7.76766528631564384965744209259, −7.59399656098393901698322032000, −6.84204304364446433100731084442, −6.65988581745657647896157723748, −5.87733638867362016013373309104, −5.51301116927304108888385171601, −4.78704165404704493206042229569, −4.27516868669716995187305897620, −3.50926715137283746047864017246, −2.88041669740428919137873956317, −2.15707309084349721067883656465, −1.38198322901071640079041464654, 0,
1.38198322901071640079041464654, 2.15707309084349721067883656465, 2.88041669740428919137873956317, 3.50926715137283746047864017246, 4.27516868669716995187305897620, 4.78704165404704493206042229569, 5.51301116927304108888385171601, 5.87733638867362016013373309104, 6.65988581745657647896157723748, 6.84204304364446433100731084442, 7.59399656098393901698322032000, 7.76766528631564384965744209259, 8.576014478069992620504614276097, 8.683059896151617221990799357979
