| L(s) = 1 | + 3.17i·5-s − 5.48·7-s − 2.48i·11-s − 5.48·13-s + 8.82i·17-s − 1.97·19-s − 38.4i·23-s + 14.9·25-s − 17.6i·29-s − 18·31-s − 17.3i·35-s − 18.4·37-s − 30.3i·41-s + 39.9·43-s − 53.3i·47-s + ⋯ |
| L(s) = 1 | + 0.634i·5-s − 0.783·7-s − 0.225i·11-s − 0.421·13-s + 0.519i·17-s − 0.103·19-s − 1.67i·23-s + 0.597·25-s − 0.608i·29-s − 0.580·31-s − 0.497i·35-s − 0.498·37-s − 0.740i·41-s + 0.928·43-s − 1.13i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9350776045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9350776045\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.17iT - 25T^{2} \) |
| 7 | \( 1 + 5.48T + 49T^{2} \) |
| 11 | \( 1 + 2.48iT - 121T^{2} \) |
| 13 | \( 1 + 5.48T + 169T^{2} \) |
| 17 | \( 1 - 8.82iT - 289T^{2} \) |
| 19 | \( 1 + 1.97T + 361T^{2} \) |
| 23 | \( 1 + 38.4iT - 529T^{2} \) |
| 29 | \( 1 + 17.6iT - 841T^{2} \) |
| 31 | \( 1 + 18T + 961T^{2} \) |
| 37 | \( 1 + 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 39.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 26.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 89.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 84.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 91.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 1.97T + 5.32e3T^{2} \) |
| 79 | \( 1 + 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 96.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 95.9T + 9.40e3T^{2} \) |
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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
−9.879696390209968317570968349196, −8.933674081983007344689757104216, −8.098585743647878571660345066670, −6.96065501351079927083006891998, −6.47759611464149719156289750149, −5.44475051265547762042478966621, −4.21491573629785285115053325603, −3.20226187790343887427544810914, −2.22613409302549191809087842174, −0.32638341885875522128353258962,
1.23049117949734598887690059642, 2.72371343904528801458088184236, 3.80240868384187541856244196741, 4.93384814371801639663250300887, 5.70172411765442369903496250520, 6.84754712568976336006457524310, 7.53469043235883728982510244969, 8.636403603174926453251961046136, 9.414772892989145771455777868657, 9.932566547595774372434073419301
