| L(s) = 1 | − 0.280·2-s + 2.67·3-s − 1.92·4-s + 5-s − 0.750·6-s − 1.90·7-s + 1.10·8-s + 4.15·9-s − 0.280·10-s − 2.92·11-s − 5.14·12-s − 5.88·13-s + 0.534·14-s + 2.67·15-s + 3.53·16-s + 2.82·17-s − 1.16·18-s − 5.60·19-s − 1.92·20-s − 5.09·21-s + 0.821·22-s + 0.527·23-s + 2.94·24-s + 25-s + 1.65·26-s + 3.09·27-s + 3.65·28-s + ⋯ |
| L(s) = 1 | − 0.198·2-s + 1.54·3-s − 0.960·4-s + 0.447·5-s − 0.306·6-s − 0.719·7-s + 0.388·8-s + 1.38·9-s − 0.0887·10-s − 0.882·11-s − 1.48·12-s − 1.63·13-s + 0.142·14-s + 0.690·15-s + 0.883·16-s + 0.685·17-s − 0.274·18-s − 1.28·19-s − 0.429·20-s − 1.11·21-s + 0.175·22-s + 0.109·23-s + 0.600·24-s + 0.200·25-s + 0.323·26-s + 0.596·27-s + 0.691·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.280T + 2T^{2} \) |
| 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 - 0.527T + 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 3.13T + 53T^{2} \) |
| 59 | \( 1 - 0.936T + 59T^{2} \) |
| 61 | \( 1 + 5.40T + 61T^{2} \) |
| 67 | \( 1 - 0.913T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 - 8.93T + 83T^{2} \) |
| 89 | \( 1 + 4.15T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−8.750232638477030438423417619046, −8.180721308345321805314012966672, −7.51774665361796675017018963603, −6.60874966685932573200679859406, −5.32030711502863979367329389910, −4.63662331530124193501552663477, −3.54397412235678492101165138574, −2.81834872532524727869920953327, −1.90389212548243491342774142333, 0,
1.90389212548243491342774142333, 2.81834872532524727869920953327, 3.54397412235678492101165138574, 4.63662331530124193501552663477, 5.32030711502863979367329389910, 6.60874966685932573200679859406, 7.51774665361796675017018963603, 8.180721308345321805314012966672, 8.750232638477030438423417619046
