| L(s) = 1 | − 0.746·2-s − 1.44·4-s + 2.56·8-s − 3·9-s + 0.969·16-s + 2.23·18-s − 3.36·23-s − 5·25-s + 9.44·29-s − 5.86·32-s + 4.32·36-s + 11.0·37-s − 1.32·43-s + 2.50·46-s + 3.73·50-s + 6.97·53-s − 7.04·58-s + 2.43·64-s − 13.8·67-s − 0.0882·71-s − 7.70·72-s − 8.26·74-s − 17.5·79-s + 9·81-s + 0.988·86-s + 4.85·92-s + 7.21·100-s + ⋯ |
| L(s) = 1 | − 0.527·2-s − 0.721·4-s + 0.908·8-s − 9-s + 0.242·16-s + 0.527·18-s − 0.700·23-s − 25-s + 1.75·29-s − 1.03·32-s + 0.721·36-s + 1.82·37-s − 0.201·43-s + 0.369·46-s + 0.527·50-s + 0.958·53-s − 0.925·58-s + 0.304·64-s − 1.69·67-s − 0.0104·71-s − 0.908·72-s − 0.960·74-s − 1.97·79-s + 81-s + 0.106·86-s + 0.505·92-s + 0.721·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.746T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 - 9.44T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 0.0882T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.949914008255016786520417729249, −7.26010787798829915891464940090, −6.15784042252014395125134819252, −5.72327460018639982704273709556, −4.73653003794560311193602967200, −4.17482230343548445094311311392, −3.19181690333186405288501560219, −2.28767304011739620727875193937, −1.06628273016809237521910760779, 0,
1.06628273016809237521910760779, 2.28767304011739620727875193937, 3.19181690333186405288501560219, 4.17482230343548445094311311392, 4.73653003794560311193602967200, 5.72327460018639982704273709556, 6.15784042252014395125134819252, 7.26010787798829915891464940090, 7.949914008255016786520417729249
