| L(s) = 1 | + 0.220·2-s + 2.81·3-s − 1.95·4-s − 0.237·5-s + 0.621·6-s + 3.12·7-s − 0.871·8-s + 4.92·9-s − 0.0524·10-s − 3.72·11-s − 5.49·12-s + 3.28·13-s + 0.690·14-s − 0.669·15-s + 3.71·16-s + 17-s + 1.08·18-s + 3.39·19-s + 0.464·20-s + 8.80·21-s − 0.820·22-s + 6.82·23-s − 2.45·24-s − 4.94·25-s + 0.724·26-s + 5.40·27-s − 6.10·28-s + ⋯ |
| L(s) = 1 | + 0.156·2-s + 1.62·3-s − 0.975·4-s − 0.106·5-s + 0.253·6-s + 1.18·7-s − 0.308·8-s + 1.64·9-s − 0.0165·10-s − 1.12·11-s − 1.58·12-s + 0.910·13-s + 0.184·14-s − 0.172·15-s + 0.927·16-s + 0.242·17-s + 0.255·18-s + 0.779·19-s + 0.103·20-s + 1.92·21-s − 0.175·22-s + 1.42·23-s − 0.500·24-s − 0.988·25-s + 0.142·26-s + 1.04·27-s − 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.566460865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566460865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.220T + 2T^{2} \) |
| 3 | \( 1 - 2.81T + 3T^{2} \) |
| 5 | \( 1 + 0.237T + 5T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 + 6.10T + 31T^{2} \) |
| 37 | \( 1 + 7.05T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 + 7.53T + 43T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 - 5.99T + 61T^{2} \) |
| 67 | \( 1 + 0.888T + 67T^{2} \) |
| 71 | \( 1 + 1.80T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 0.985T + 89T^{2} \) |
| 97 | \( 1 - 8.38T + 97T^{2} \) |
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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
−10.06350762516520317524779150478, −9.162742903655587638837413167524, −8.466527598380119347036932823515, −8.052898070790886915631453394142, −7.26080200511501957251146025444, −5.50735137818451923521308310887, −4.73935386323084077591677685203, −3.70695754975585095703368432858, −2.86790250492502009520962459174, −1.43702423232293269575149011273,
1.43702423232293269575149011273, 2.86790250492502009520962459174, 3.70695754975585095703368432858, 4.73935386323084077591677685203, 5.50735137818451923521308310887, 7.26080200511501957251146025444, 8.052898070790886915631453394142, 8.466527598380119347036932823515, 9.162742903655587638837413167524, 10.06350762516520317524779150478
