| L(s) = 1 | − 1.14·2-s + 2.15·3-s − 0.679·4-s − 3.87·5-s − 2.47·6-s + 3.07·8-s + 1.63·9-s + 4.45·10-s + 11-s − 1.46·12-s + 4.09·13-s − 8.34·15-s − 2.17·16-s − 0.824·17-s − 1.87·18-s + 4.50·19-s + 2.63·20-s − 1.14·22-s + 4.86·23-s + 6.62·24-s + 10.0·25-s − 4.70·26-s − 2.93·27-s + 7.79·29-s + 9.58·30-s + 3.82·31-s − 3.65·32-s + ⋯ |
| L(s) = 1 | − 0.812·2-s + 1.24·3-s − 0.339·4-s − 1.73·5-s − 1.00·6-s + 1.08·8-s + 0.545·9-s + 1.40·10-s + 0.301·11-s − 0.422·12-s + 1.13·13-s − 2.15·15-s − 0.544·16-s − 0.200·17-s − 0.442·18-s + 1.03·19-s + 0.589·20-s − 0.244·22-s + 1.01·23-s + 1.35·24-s + 2.00·25-s − 0.923·26-s − 0.565·27-s + 1.44·29-s + 1.75·30-s + 0.687·31-s − 0.646·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9949832666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9949832666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 + 0.824T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 0.868T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 + 2.10T + 97T^{2} \) |
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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
−10.77040432389710391620973397573, −9.580932193587416235084721030743, −8.821369704240671949748170844847, −8.227042858684009862477163536500, −7.78028242034006832696373651864, −6.76668549939965936196760312910, −4.80502535312923754094096816275, −3.85828791254360964695939080971, −3.07584986966519302430491849397, −1.01383440042592199891709032449,
1.01383440042592199891709032449, 3.07584986966519302430491849397, 3.85828791254360964695939080971, 4.80502535312923754094096816275, 6.76668549939965936196760312910, 7.78028242034006832696373651864, 8.227042858684009862477163536500, 8.821369704240671949748170844847, 9.580932193587416235084721030743, 10.77040432389710391620973397573
