| L(s) = 1 | − 1.07i·5-s + 7.21·7-s + 16.3i·11-s + 21.6·13-s + 18.9i·17-s + 17.0·19-s + 1.11i·23-s + 23.8·25-s − 29.4i·29-s − 5.63·31-s − 7.75i·35-s − 17.0·37-s + 27.4i·41-s − 52.3·43-s + 64.5i·47-s + ⋯ |
| L(s) = 1 | − 0.214i·5-s + 1.03·7-s + 1.48i·11-s + 1.66·13-s + 1.11i·17-s + 0.897·19-s + 0.0485i·23-s + 0.953·25-s − 1.01i·29-s − 0.181·31-s − 0.221i·35-s − 0.460·37-s + 0.669i·41-s − 1.21·43-s + 1.37i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.647291443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647291443\) |
| \(L(2)\) |
|
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 | \( 1 + 1.07iT - 25T^{2} \) |
| 7 | \( 1 - 7.21T + 49T^{2} \) |
| 11 | \( 1 - 16.3iT - 121T^{2} \) |
| 13 | \( 1 - 21.6T + 169T^{2} \) |
| 17 | \( 1 - 18.9iT - 289T^{2} \) |
| 19 | \( 1 - 17.0T + 361T^{2} \) |
| 23 | \( 1 - 1.11iT - 529T^{2} \) |
| 29 | \( 1 + 29.4iT - 841T^{2} \) |
| 31 | \( 1 + 5.63T + 961T^{2} \) |
| 37 | \( 1 + 17.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 52.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 56.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 69.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 98.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 7.75iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 76.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.84T + 9.40e3T^{2} \) |
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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
−8.836587082878526653950811142296, −8.155308854730520738695043915378, −7.57064917518919126610922382190, −6.58623301344131398595332150032, −5.81286084893041379957916968636, −4.82574071054270874144225552558, −4.26923124679266477305336703500, −3.22672599309104584992815242368, −1.81899064377171333151863695770, −1.25041416262631540497005604027,
0.71786232759551217894640169585, 1.60734898557946565666905116229, 3.08677628830640294213885569231, 3.58447835358054931255767912615, 4.89821715316694723583477459106, 5.46926589246968852738875968566, 6.36375651558407065038204823589, 7.17944033411131598795651967950, 8.082548169524311211760928552699, 8.701668063283343104678397484997
