| L(s) = 1 | + 3-s + 4·5-s + 9-s + 4·15-s + 8·19-s − 16·23-s + 2·25-s + 27-s − 12·29-s − 8·43-s + 4·45-s − 14·49-s + 4·53-s + 8·57-s + 8·67-s − 16·69-s + 16·71-s + 20·73-s + 2·75-s + 81-s − 12·87-s + 32·95-s + 4·97-s + 36·101-s − 64·115-s − 6·121-s − 28·125-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.03·15-s + 1.83·19-s − 3.33·23-s + 2/5·25-s + 0.192·27-s − 2.22·29-s − 1.21·43-s + 0.596·45-s − 2·49-s + 0.549·53-s + 1.05·57-s + 0.977·67-s − 1.92·69-s + 1.89·71-s + 2.34·73-s + 0.230·75-s + 1/9·81-s − 1.28·87-s + 3.28·95-s + 0.406·97-s + 3.58·101-s − 5.96·115-s − 0.545·121-s − 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898583062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898583062\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.20677110069933956172337352464, −9.785656483951954927896383611288, −9.747086677582411849557090474382, −9.298977106556460794427243537385, −8.456614023795656741730916500766, −7.77541900804615864403738320707, −7.61947548271568794045481490015, −6.50812189570177474416634496690, −6.16826675668887144123811676692, −5.48498394296229820428473072147, −5.14444399128618341481533912587, −3.89741909491206350829507060388, −3.43299575130251299450107440988, −1.99999348241168201324552330088, −1.97363471421369880359461961552,
1.97363471421369880359461961552, 1.99999348241168201324552330088, 3.43299575130251299450107440988, 3.89741909491206350829507060388, 5.14444399128618341481533912587, 5.48498394296229820428473072147, 6.16826675668887144123811676692, 6.50812189570177474416634496690, 7.61947548271568794045481490015, 7.77541900804615864403738320707, 8.456614023795656741730916500766, 9.298977106556460794427243537385, 9.747086677582411849557090474382, 9.785656483951954927896383611288, 10.20677110069933956172337352464
