| L(s) = 1 | − 7-s − 4·9-s + 2·11-s + 10·23-s + 2·25-s − 16·29-s + 4·37-s + 10·43-s + 49-s − 10·53-s + 4·63-s − 2·77-s − 20·79-s + 7·81-s − 8·99-s + 4·107-s + 16·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 10·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 4/3·9-s + 0.603·11-s + 2.08·23-s + 2/5·25-s − 2.97·29-s + 0.657·37-s + 1.52·43-s + 1/7·49-s − 1.37·53-s + 0.503·63-s − 0.227·77-s − 2.25·79-s + 7/9·81-s − 0.804·99-s + 0.386·107-s + 1.53·109-s − 1.63·121-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 − 0.788·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
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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
−7.63730866217015373707714422514, −7.30938610443441778756699473080, −6.92786456894508405007857583570, −6.36249529789059442871028676114, −5.92419077433525637228473259977, −5.60186678453625226372964075247, −5.18685509698218462461608229094, −4.59760253603988668246969274775, −4.03608431046928950244930672346, −3.49808653690208959590256103111, −3.02489626653297184699187880187, −2.61942115191895265346521724222, −1.83244092276388395722828423858, −1.02146009644930970224782609679, 0,
1.02146009644930970224782609679, 1.83244092276388395722828423858, 2.61942115191895265346521724222, 3.02489626653297184699187880187, 3.49808653690208959590256103111, 4.03608431046928950244930672346, 4.59760253603988668246969274775, 5.18685509698218462461608229094, 5.60186678453625226372964075247, 5.92419077433525637228473259977, 6.36249529789059442871028676114, 6.92786456894508405007857583570, 7.30938610443441778756699473080, 7.63730866217015373707714422514
