| L(s) = 1 | − 0.150·2-s + 1.96·3-s − 1.97·4-s − 0.296·6-s + 1.54·7-s + 0.600·8-s + 0.849·9-s + 4.56·11-s − 3.87·12-s + 1.09·13-s − 0.233·14-s + 3.86·16-s − 17-s − 0.128·18-s + 4.67·19-s + 3.03·21-s − 0.688·22-s − 0.529·23-s + 1.17·24-s − 0.165·26-s − 4.21·27-s − 3.05·28-s + 8.06·29-s − 4.78·31-s − 1.78·32-s + 8.94·33-s + 0.150·34-s + ⋯ |
| L(s) = 1 | − 0.106·2-s + 1.13·3-s − 0.988·4-s − 0.120·6-s + 0.583·7-s + 0.212·8-s + 0.283·9-s + 1.37·11-s − 1.11·12-s + 0.304·13-s − 0.0623·14-s + 0.965·16-s − 0.242·17-s − 0.0302·18-s + 1.07·19-s + 0.661·21-s − 0.146·22-s − 0.110·23-s + 0.240·24-s − 0.0324·26-s − 0.812·27-s − 0.577·28-s + 1.49·29-s − 0.858·31-s − 0.315·32-s + 1.55·33-s + 0.0258·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706147333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706147333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.150T + 2T^{2} \) |
| 3 | \( 1 - 1.96T + 3T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + 0.529T + 23T^{2} \) |
| 29 | \( 1 - 8.06T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 + 0.751T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0227T + 53T^{2} \) |
| 59 | \( 1 + 3.56T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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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
−11.22382625612168569055376572585, −9.837117562961723490156960122790, −9.290956628623879541658869299029, −8.461109531141068427812045706258, −7.938228224848120325542695626842, −6.60943714804173446178976861842, −5.19883094514560799234661131307, −4.11341002380124242175242076680, −3.20351669578278578364384614033, −1.45571908535926902852269873139,
1.45571908535926902852269873139, 3.20351669578278578364384614033, 4.11341002380124242175242076680, 5.19883094514560799234661131307, 6.60943714804173446178976861842, 7.938228224848120325542695626842, 8.461109531141068427812045706258, 9.290956628623879541658869299029, 9.837117562961723490156960122790, 11.22382625612168569055376572585
