| L(s) = 1 | − 2i·2-s + 5.86i·3-s − 4·4-s + 11.7·6-s + 11.5i·7-s + 8i·8-s − 25.3·9-s − 23.4i·12-s + 23.0·14-s + 16·16-s + 50.7i·18-s − 67.5·21-s − 0.836i·23-s − 46.9·24-s − 96.1i·27-s − 46.0i·28-s + ⋯ |
| L(s) = 1 | − i·2-s + 1.95i·3-s − 4-s + 1.95·6-s + 1.64i·7-s + i·8-s − 2.82·9-s − 1.95i·12-s + 1.64·14-s + 16-s + 2.82i·18-s − 3.21·21-s − 0.0363i·23-s − 1.95·24-s − 3.56i·27-s − 1.64i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7680333945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7680333945\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.86iT - 9T^{2} \) |
| 7 | \( 1 - 11.5iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 0.836iT - 529T^{2} \) |
| 29 | \( 1 - 13.7T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 85.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 120.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 116iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 163. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−11.04650733251271555909820939129, −10.27376969450182135855296678713, −9.490945161929961522069144790877, −8.899114353407860005741346249816, −8.308634629048341309574768836209, −5.89422172068460007176162809163, −5.24614747999899342243627945614, −4.37079370710940540170687294058, −3.27284123856378812482892867330, −2.43006486441150104886353687500,
0.32373286402654233738429927434, 1.39734707631635743986303475938, 3.30742242109628105610240170296, 4.77187977228498809286718629777, 6.11926924584110768556967147404, 6.71971586173339008070264811325, 7.52112302854755543824623332958, 7.929483447185602149220048190959, 8.949343753530252986816559343735, 10.23947940906484625891942500810
