| L(s) = 1 | − 1.53·2-s + 2.87·3-s + 0.347·4-s + 2.34·5-s − 4.41·6-s + 2.53·8-s + 5.29·9-s − 3.59·10-s + 11-s + 0.999·12-s − 0.184·13-s + 6.75·15-s − 4.57·16-s − 3.92·17-s − 8.10·18-s + 0.773·19-s + 0.815·20-s − 1.53·22-s + 8.35·23-s + 7.29·24-s + 0.509·25-s + 0.283·26-s + 6.59·27-s − 8.17·29-s − 10.3·30-s + 2.65·31-s + 1.94·32-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 1.66·3-s + 0.173·4-s + 1.04·5-s − 1.80·6-s + 0.895·8-s + 1.76·9-s − 1.13·10-s + 0.301·11-s + 0.288·12-s − 0.0512·13-s + 1.74·15-s − 1.14·16-s − 0.951·17-s − 1.91·18-s + 0.177·19-s + 0.182·20-s − 0.326·22-s + 1.74·23-s + 1.48·24-s + 0.101·25-s + 0.0555·26-s + 1.26·27-s − 1.51·29-s − 1.89·30-s + 0.476·31-s + 0.343·32-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
| \(L(1)\) |
\(\approx\) |
\(1.639791381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.639791381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 13 | \( 1 + 0.184T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 - 0.773T + 19T^{2} \) |
| 23 | \( 1 - 8.35T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 0.426T + 41T^{2} \) |
| 43 | \( 1 - 1.18T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 - 0.204T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 3.75T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 - 0.120T + 73T^{2} \) |
| 79 | \( 1 - 0.327T + 79T^{2} \) |
| 83 | \( 1 - 3.35T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−10.30427717192300904519517314222, −9.590724619340757239077753687657, −9.030285844044607041270207343199, −8.561105138671921394763248532186, −7.50545284129951312740717821613, −6.75088905224832820031770971635, −5.11985004261060567397917744596, −3.82474645514583597149794132363, −2.47373480484561090507506086664, −1.54280918184852114445669431461,
1.54280918184852114445669431461, 2.47373480484561090507506086664, 3.82474645514583597149794132363, 5.11985004261060567397917744596, 6.75088905224832820031770971635, 7.50545284129951312740717821613, 8.561105138671921394763248532186, 9.030285844044607041270207343199, 9.590724619340757239077753687657, 10.30427717192300904519517314222
