| L(s) = 1 | − 2·7-s + 4·9-s − 2·11-s − 6·25-s − 12·29-s − 6·37-s − 3·49-s − 18·53-s − 8·63-s + 2·67-s + 18·71-s + 4·77-s + 4·79-s + 7·81-s − 8·99-s − 22·107-s − 10·109-s − 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 4/3·9-s − 0.603·11-s − 6/5·25-s − 2.22·29-s − 0.986·37-s − 3/7·49-s − 2.47·53-s − 1.00·63-s + 0.244·67-s + 2.13·71-s + 0.455·77-s + 0.450·79-s + 7/9·81-s − 0.804·99-s − 2.12·107-s − 0.957·109-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
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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
−9.201817625931730631319801866861, −8.231653912108904502987139381029, −7.85913569995824361155632140743, −7.55872617707535023036071879644, −6.81021046990522680701216121139, −6.66875431209654207875868431102, −5.92283422217353581047274630014, −5.39740932329437370318814573989, −4.91206401871728991161913348225, −4.13475868340641532676358143122, −3.71743918797515679579241650492, −3.15247985928906233338458747554, −2.15777317103253069988565475152, −1.56577411252041071390112849468, 0,
1.56577411252041071390112849468, 2.15777317103253069988565475152, 3.15247985928906233338458747554, 3.71743918797515679579241650492, 4.13475868340641532676358143122, 4.91206401871728991161913348225, 5.39740932329437370318814573989, 5.92283422217353581047274630014, 6.66875431209654207875868431102, 6.81021046990522680701216121139, 7.55872617707535023036071879644, 7.85913569995824361155632140743, 8.231653912108904502987139381029, 9.201817625931730631319801866861
