| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.95i·5-s − 2.64·7-s + 2.82i·8-s + 4.18·10-s − 2.24i·11-s − 4.48·13-s + 3.74i·14-s + 4.00·16-s + 3.17i·17-s + 15.1·19-s − 5.91i·20-s − 3.17·22-s − 20.9i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.591i·5-s − 0.377·7-s + 0.353i·8-s + 0.418·10-s − 0.203i·11-s − 0.344·13-s + 0.267i·14-s + 0.250·16-s + 0.186i·17-s + 0.795·19-s − 0.295i·20-s − 0.144·22-s − 0.909i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4616339891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4616339891\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 2.95iT - 25T^{2} \) |
| 11 | \( 1 + 2.24iT - 121T^{2} \) |
| 13 | \( 1 + 4.48T + 169T^{2} \) |
| 17 | \( 1 - 3.17iT - 289T^{2} \) |
| 19 | \( 1 - 15.1T + 361T^{2} \) |
| 23 | \( 1 + 20.9iT - 529T^{2} \) |
| 29 | \( 1 - 33.3iT - 841T^{2} \) |
| 31 | \( 1 + 51.5T + 961T^{2} \) |
| 37 | \( 1 + 8.74T + 1.36e3T^{2} \) |
| 41 | \( 1 + 70.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 44.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 18.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 38.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 73.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 23.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 78.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 164. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.7T + 9.40e3T^{2} \) |
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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
−9.251214946632977069249430360339, −8.677005188597303768169123874601, −7.48185422800739197324391902847, −6.80528075568621543014686146352, −5.70732376536551625295936221514, −4.80143070300982335828753624836, −3.57702428889994740990161318142, −2.93220856170476965847476624603, −1.71126326249944533204132955681, −0.14313336148252000435020315851,
1.34879571912087809207120243146, 2.97077265659233561071741256143, 4.10606246571820651329188924892, 5.06785936735594935637220032038, 5.74203267998398573291879163390, 6.78883093272041595415710746332, 7.53186390449518446080450331026, 8.289138576938807682983744927635, 9.353627252310415799466412480950, 9.594490526805809426775459710369
