| L(s) = 1 | + 3-s − 4·5-s + 9-s − 4·15-s + 11·25-s + 27-s − 8·31-s − 4·45-s − 2·49-s − 24·53-s + 11·75-s + 24·79-s + 81-s − 8·83-s − 8·93-s − 24·107-s + 6·121-s − 24·125-s + 127-s + 131-s − 4·135-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 32·155-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.03·15-s + 11/5·25-s + 0.192·27-s − 1.43·31-s − 0.596·45-s − 2/7·49-s − 3.29·53-s + 1.27·75-s + 2.70·79-s + 1/9·81-s − 0.878·83-s − 0.829·93-s − 2.32·107-s + 6/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s − 0.344·135-s + 0.0854·137-s + 0.0848·139-s − 0.164·147-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.911200952565872692692679267299, −8.268416427686795174252954344681, −8.072341552149252345439757261876, −7.56436560186957102598739279953, −7.23359338659161847548622275353, −6.62911343606471084787024441778, −6.14321473041312097454251796019, −5.21384735238755137785851563206, −4.79349830645196424720465892089, −4.20533247699690032625598387341, −3.57688370869089639619151797451, −3.31964777505009377642066877102, −2.46709284284166354256118350374, −1.40801627514083385851807859549, 0,
1.40801627514083385851807859549, 2.46709284284166354256118350374, 3.31964777505009377642066877102, 3.57688370869089639619151797451, 4.20533247699690032625598387341, 4.79349830645196424720465892089, 5.21384735238755137785851563206, 6.14321473041312097454251796019, 6.62911343606471084787024441778, 7.23359338659161847548622275353, 7.56436560186957102598739279953, 8.072341552149252345439757261876, 8.268416427686795174252954344681, 8.911200952565872692692679267299
