| L(s) = 1 | + 0.445·2-s − 0.801·4-s + 1.80·5-s − 0.801·8-s + 0.801·10-s + 0.445·16-s − 0.445·19-s − 1.44·20-s + 2.24·25-s − 1.24·29-s + 32-s − 0.445·37-s − 0.198·38-s − 1.44·40-s − 1.80·43-s + 49-s + 50-s − 0.554·58-s − 71-s − 1.80·73-s − 0.198·74-s + 0.356·76-s − 1.80·79-s + 0.801·80-s + 0.445·83-s − 0.801·86-s − 1.24·89-s + ⋯ |
| L(s) = 1 | + 0.445·2-s − 0.801·4-s + 1.80·5-s − 0.801·8-s + 0.801·10-s + 0.445·16-s − 0.445·19-s − 1.44·20-s + 2.24·25-s − 1.24·29-s + 32-s − 0.445·37-s − 0.198·38-s − 1.44·40-s − 1.80·43-s + 49-s + 50-s − 0.554·58-s − 71-s − 1.80·73-s − 0.198·74-s + 0.356·76-s − 1.80·79-s + 0.801·80-s + 0.445·83-s − 0.801·86-s − 1.24·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.207028707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207028707\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 0.445T + T^{2} \) |
| 5 | \( 1 - 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 - 0.445T + T^{2} \) |
| 89 | \( 1 + 1.24T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50459529394021378671043971772, −9.889211950552135803505378522995, −9.146777271039715882952195593566, −8.486252079886083166878407002878, −7.01065521827902659158644931458, −5.94983306036910194011291288850, −5.44912832043962092066970112486, −4.43567647761754511569812735999, −3.08842507256238758207915794763, −1.77073079791246240609274844577,
1.77073079791246240609274844577, 3.08842507256238758207915794763, 4.43567647761754511569812735999, 5.44912832043962092066970112486, 5.94983306036910194011291288850, 7.01065521827902659158644931458, 8.486252079886083166878407002878, 9.146777271039715882952195593566, 9.889211950552135803505378522995, 10.50459529394021378671043971772
