| L(s) = 1 | + 1.55·2-s + 3.00·3-s + 0.419·4-s + 4.67·6-s + 3.45·7-s − 2.45·8-s + 6.04·9-s − 4.24·11-s + 1.26·12-s + 0.127·13-s + 5.37·14-s − 4.66·16-s + 9.40·18-s + 2.57·19-s + 10.3·21-s − 6.60·22-s + 3.25·23-s − 7.39·24-s + 0.198·26-s + 9.16·27-s + 1.45·28-s + 4.90·29-s + 5.36·31-s − 2.33·32-s − 12.7·33-s + 2.53·36-s − 1.77·37-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 1.73·3-s + 0.209·4-s + 1.91·6-s + 1.30·7-s − 0.869·8-s + 2.01·9-s − 1.27·11-s + 0.364·12-s + 0.0353·13-s + 1.43·14-s − 1.16·16-s + 2.21·18-s + 0.590·19-s + 2.26·21-s − 1.40·22-s + 0.678·23-s − 1.50·24-s + 0.0388·26-s + 1.76·27-s + 0.274·28-s + 0.910·29-s + 0.963·31-s − 0.413·32-s − 2.22·33-s + 0.423·36-s − 0.291·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.324041159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.324041159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 3 | \( 1 - 3.00T + 3T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 0.127T + 13T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 + 6.88T + 67T^{2} \) |
| 71 | \( 1 - 7.21T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 0.600T + 89T^{2} \) |
| 97 | \( 1 + 7.67T + 97T^{2} \) |
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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
−8.056117926749813112055519604486, −7.42555203049737932633355533612, −6.56548968180410082153144678525, −5.41333216190882544700001483697, −4.94905608637436466509812207478, −4.29741712510131891551712398385, −3.57574004808655808331314637640, −2.66772828812542594055698271072, −2.42890876426294951906838977338, −1.14080106219177264183217775226,
1.14080106219177264183217775226, 2.42890876426294951906838977338, 2.66772828812542594055698271072, 3.57574004808655808331314637640, 4.29741712510131891551712398385, 4.94905608637436466509812207478, 5.41333216190882544700001483697, 6.56548968180410082153144678525, 7.42555203049737932633355533612, 8.056117926749813112055519604486
