| L(s) = 1 | + 1.39·2-s + 0.475·3-s − 0.0485·4-s + 5-s + 0.664·6-s − 3.39·7-s − 2.86·8-s − 2.77·9-s + 1.39·10-s − 3.53·11-s − 0.0231·12-s + 3.99·13-s − 4.73·14-s + 0.475·15-s − 3.90·16-s − 2.63·17-s − 3.87·18-s − 1.70·19-s − 0.0485·20-s − 1.61·21-s − 4.93·22-s − 6.78·23-s − 1.36·24-s + 25-s + 5.57·26-s − 2.74·27-s + 0.164·28-s + ⋯ |
| L(s) = 1 | + 0.987·2-s + 0.274·3-s − 0.0242·4-s + 0.447·5-s + 0.271·6-s − 1.28·7-s − 1.01·8-s − 0.924·9-s + 0.441·10-s − 1.06·11-s − 0.00667·12-s + 1.10·13-s − 1.26·14-s + 0.122·15-s − 0.975·16-s − 0.638·17-s − 0.913·18-s − 0.390·19-s − 0.0108·20-s − 0.352·21-s − 1.05·22-s − 1.41·23-s − 0.278·24-s + 0.200·25-s + 1.09·26-s − 0.528·27-s + 0.0311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 - 0.475T + 3T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 + 6.78T + 23T^{2} \) |
| 29 | \( 1 - 0.702T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 4.90T + 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 4.51T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 - 6.80T + 73T^{2} \) |
| 79 | \( 1 - 0.211T + 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 4.38T + 97T^{2} \) |
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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
−9.615466405275665711187589009060, −8.704593581418064253330122071506, −8.116453292705037966615003939403, −6.46164747611514702736494907588, −6.14608203324834233044331255285, −5.27401989315577270301185570702, −4.12590564986601276572006795203, −3.19527127890179035681248992515, −2.47756056747767191796496656235, 0,
2.47756056747767191796496656235, 3.19527127890179035681248992515, 4.12590564986601276572006795203, 5.27401989315577270301185570702, 6.14608203324834233044331255285, 6.46164747611514702736494907588, 8.116453292705037966615003939403, 8.704593581418064253330122071506, 9.615466405275665711187589009060
