| L(s) = 1 | − 2-s + 4-s − 8-s − 4·9-s + 16-s + 4·17-s + 4·18-s − 10·23-s − 25-s + 14·31-s − 32-s − 4·34-s − 4·36-s − 10·41-s + 10·46-s − 6·47-s + 49-s + 50-s − 14·62-s + 64-s + 4·68-s − 6·71-s + 4·72-s − 6·73-s − 6·79-s + 7·81-s + 10·82-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 4/3·9-s + 1/4·16-s + 0.970·17-s + 0.942·18-s − 2.08·23-s − 1/5·25-s + 2.51·31-s − 0.176·32-s − 0.685·34-s − 2/3·36-s − 1.56·41-s + 1.47·46-s − 0.875·47-s + 1/7·49-s + 0.141·50-s − 1.77·62-s + 1/8·64-s + 0.485·68-s − 0.712·71-s + 0.471·72-s − 0.702·73-s − 0.675·79-s + 7/9·81-s + 1.10·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.938998706771454137084942232287, −8.365273693022602986215946355306, −8.198086352970581889519039050869, −7.82189625633500615773206421173, −7.16043449295330893191987730903, −6.40897691815213503359906651675, −6.18098062514471055090955144662, −5.63893510615523506254116933186, −5.06838892300269114635217932760, −4.33024742887391667749911118531, −3.56597258867792779196389690253, −2.95600844045260858276769526124, −2.35472488539213125000069914777, −1.37117321424817614837017968493, 0,
1.37117321424817614837017968493, 2.35472488539213125000069914777, 2.95600844045260858276769526124, 3.56597258867792779196389690253, 4.33024742887391667749911118531, 5.06838892300269114635217932760, 5.63893510615523506254116933186, 6.18098062514471055090955144662, 6.40897691815213503359906651675, 7.16043449295330893191987730903, 7.82189625633500615773206421173, 8.198086352970581889519039050869, 8.365273693022602986215946355306, 8.938998706771454137084942232287
