| L(s) = 1 | + 2·2-s + 2·4-s − 4·5-s − 3·9-s − 8·10-s − 4·16-s + 4·17-s − 6·18-s − 4·19-s − 8·20-s + 3·25-s − 12·31-s − 8·32-s + 8·34-s − 6·36-s − 8·38-s + 12·45-s − 5·49-s + 6·50-s − 4·59-s − 6·61-s − 24·62-s − 8·64-s + 8·67-s + 8·68-s + 20·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.78·5-s − 9-s − 2.52·10-s − 16-s + 0.970·17-s − 1.41·18-s − 0.917·19-s − 1.78·20-s + 3/5·25-s − 2.15·31-s − 1.41·32-s + 1.37·34-s − 36-s − 1.29·38-s + 1.78·45-s − 5/7·49-s + 0.848·50-s − 0.520·59-s − 0.768·61-s − 3.04·62-s − 64-s + 0.977·67-s + 0.970·68-s + 2.37·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.840360787216033746389997669918, −9.163854534112985864093296914038, −8.718458811739825938015906211663, −8.099827964219928532254689747376, −7.68496073909801128060750366511, −7.24515174833988262333241601995, −6.40254997614512219504271199814, −6.01672290050992865612394817495, −5.23529369529126727124225694545, −4.90160075418881548020588288372, −3.90189243844002528726442717108, −3.76086462527045399654604291734, −3.17449913989658698253191092446, −2.19320558662792427408159197570, 0,
2.19320558662792427408159197570, 3.17449913989658698253191092446, 3.76086462527045399654604291734, 3.90189243844002528726442717108, 4.90160075418881548020588288372, 5.23529369529126727124225694545, 6.01672290050992865612394817495, 6.40254997614512219504271199814, 7.24515174833988262333241601995, 7.68496073909801128060750366511, 8.099827964219928532254689747376, 8.718458811739825938015906211663, 9.163854534112985864093296914038, 9.840360787216033746389997669918
