| L(s) = 1 | + (4.52 − 2.12i)5-s − 0.343i·7-s + 3.68i·11-s − 5.50i·13-s + 6.36·17-s + 2.62·19-s − 12.2·23-s + (15.9 − 19.2i)25-s − 24.0i·29-s + 1.04·31-s + (−0.729 − 1.55i)35-s − 36.2i·37-s + 8.06i·41-s + 19.3i·43-s + 38.7·47-s + ⋯ |
| L(s) = 1 | + (0.905 − 0.425i)5-s − 0.0490i·7-s + 0.334i·11-s − 0.423i·13-s + 0.374·17-s + 0.138·19-s − 0.533·23-s + (0.638 − 0.769i)25-s − 0.828i·29-s + 0.0337·31-s + (−0.0208 − 0.0443i)35-s − 0.979i·37-s + 0.196i·41-s + 0.449i·43-s + 0.824·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.339662592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.339662592\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.52 + 2.12i)T \) |
| good | 7 | \( 1 + 0.343iT - 49T^{2} \) |
| 11 | \( 1 - 3.68iT - 121T^{2} \) |
| 13 | \( 1 + 5.50iT - 169T^{2} \) |
| 17 | \( 1 - 6.36T + 289T^{2} \) |
| 19 | \( 1 - 2.62T + 361T^{2} \) |
| 23 | \( 1 + 12.2T + 529T^{2} \) |
| 29 | \( 1 + 24.0iT - 841T^{2} \) |
| 31 | \( 1 - 1.04T + 961T^{2} \) |
| 37 | \( 1 + 36.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.06iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 3.24T + 2.80e3T^{2} \) |
| 59 | \( 1 + 51.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 77.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 108. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 78.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 35.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 65.0iT - 9.40e3T^{2} \) |
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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
−8.849519825282976006781134323337, −7.996194158688117391799364227753, −7.25514227567987444353365206764, −6.21792264960395147356071660227, −5.65691641120240140700293742054, −4.81939800031024998323921807132, −3.89665815126146577486180656671, −2.69657455309781269778047474724, −1.80147428834616529069564419399, −0.62059141102165334455613256438,
1.12940376088264433790456759508, 2.20327428346451379633958785336, 3.10659549353048020388584107784, 4.10939583642853321779679211717, 5.25119821181268042392866551484, 5.84338736633590291479134862546, 6.69513488283933821665759503446, 7.33429523527432624177834362891, 8.387772240300479727383083834653, 9.058661210836890141999244815746
