| L(s) = 1 | + 9-s + 4·11-s − 4·16-s − 10·19-s + 20·29-s − 6·31-s − 16·41-s − 5·49-s − 20·59-s + 14·61-s − 16·71-s + 81-s + 4·99-s + 24·101-s + 10·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.20·11-s − 16-s − 2.29·19-s + 3.71·29-s − 1.07·31-s − 2.49·41-s − 5/7·49-s − 2.60·59-s + 1.79·61-s − 1.89·71-s + 1/9·81-s + 0.402·99-s + 2.38·101-s + 0.957·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8797220405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8797220405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−12.06535678251561534347450688093, −11.69351541999948731726714975337, −10.88974094536361104567387978062, −10.39882094959826449711333942129, −9.903619965777680376182050580499, −8.979440455637625961269202062502, −8.655074250265874594654667643843, −8.114757177440921419069207299350, −6.79375443883521821659151920822, −6.77287251257420576381216593643, −6.05569215012514529764032801313, −4.62309319523854591570832526017, −4.48018036005395779311676580836, −3.24466149791278329969031792942, −1.90319955654708104595670502735,
1.90319955654708104595670502735, 3.24466149791278329969031792942, 4.48018036005395779311676580836, 4.62309319523854591570832526017, 6.05569215012514529764032801313, 6.77287251257420576381216593643, 6.79375443883521821659151920822, 8.114757177440921419069207299350, 8.655074250265874594654667643843, 8.979440455637625961269202062502, 9.903619965777680376182050580499, 10.39882094959826449711333942129, 10.88974094536361104567387978062, 11.69351541999948731726714975337, 12.06535678251561534347450688093
