| L(s) = 1 | + 2·5-s − 2·9-s − 4·11-s − 12·19-s − 25-s − 4·29-s − 4·41-s − 4·45-s + 6·49-s − 8·55-s − 4·59-s − 12·61-s − 8·71-s + 16·79-s − 5·81-s + 4·89-s − 24·95-s + 8·99-s + 12·101-s + 20·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2/3·9-s − 1.20·11-s − 2.75·19-s − 1/5·25-s − 0.742·29-s − 0.624·41-s − 0.596·45-s + 6/7·49-s − 1.07·55-s − 0.520·59-s − 1.53·61-s − 0.949·71-s + 1.80·79-s − 5/9·81-s + 0.423·89-s − 2.46·95-s + 0.804·99-s + 1.19·101-s + 1.91·109-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
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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
−9.053741841487707353882126684056, −8.990015121740525383648379496576, −8.373164867544173382186980638339, −7.87156441916130722244009057393, −7.41031075946135687926852450580, −6.61127426785943061529500519900, −6.15692190774184236929415211667, −5.83117384836379108269028934953, −5.18695218824486532825846447060, −4.62582934312640197935671693414, −3.95524337065984516997232290540, −3.07982062010987887626317506271, −2.32406872197087040622380442993, −1.89724448843746106184694344043, 0,
1.89724448843746106184694344043, 2.32406872197087040622380442993, 3.07982062010987887626317506271, 3.95524337065984516997232290540, 4.62582934312640197935671693414, 5.18695218824486532825846447060, 5.83117384836379108269028934953, 6.15692190774184236929415211667, 6.61127426785943061529500519900, 7.41031075946135687926852450580, 7.87156441916130722244009057393, 8.373164867544173382186980638339, 8.990015121740525383648379496576, 9.053741841487707353882126684056
