| L(s) = 1 | + 4.76·3-s + (1.82 − 4.65i)5-s + (1.46 + 6.84i)7-s + 13.6·9-s + 16.6·11-s − 14.9·13-s + (8.67 − 22.1i)15-s + 11.3·17-s − 3.16i·19-s + (7 + 32.5i)21-s + 22.1i·23-s + (−18.3 − 16.9i)25-s + 22.2·27-s + 16.0·29-s − 21.7i·31-s + ⋯ |
| L(s) = 1 | + 1.58·3-s + (0.364 − 0.931i)5-s + (0.209 + 0.977i)7-s + 1.51·9-s + 1.51·11-s − 1.15·13-s + (0.578 − 1.47i)15-s + 0.667·17-s − 0.166i·19-s + (0.333 + 1.55i)21-s + 0.964i·23-s + (−0.734 − 0.678i)25-s + 0.825·27-s + 0.552·29-s − 0.702i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.509509613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.509509613\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.82 + 4.65i)T \) |
| 7 | \( 1 + (-1.46 - 6.84i)T \) |
| good | 3 | \( 1 - 4.76T + 9T^{2} \) |
| 11 | \( 1 - 16.6T + 121T^{2} \) |
| 13 | \( 1 + 14.9T + 169T^{2} \) |
| 17 | \( 1 - 11.3T + 289T^{2} \) |
| 19 | \( 1 + 3.16iT - 361T^{2} \) |
| 23 | \( 1 - 22.1iT - 529T^{2} \) |
| 29 | \( 1 - 16.0T + 841T^{2} \) |
| 31 | \( 1 + 21.7iT - 961T^{2} \) |
| 37 | \( 1 - 35.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 74.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 11.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 20.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 93.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 13.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 40.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 40.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 42.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 52.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 116.T + 9.40e3T^{2} \) |
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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
−10.01229574222229807828705178351, −9.344696182491248259100821222071, −8.921281575837982357414865991401, −8.135178150274812725223490344165, −7.20671245928795821266789762790, −5.86020905703823907237005014681, −4.76335839809557552893636970337, −3.66904510035852343946698654876, −2.45396077732985498326422597150, −1.48962557351781215925340272848,
1.51394952944731141243128296989, 2.74757974257565429097217060447, 3.61316071600094076513866914311, 4.56429051076050824932484256740, 6.37886752336289693548622519695, 7.16941975039958598599366054485, 7.83006365888103933525078513122, 8.866836044190504185346099385284, 9.764883967348713102155395149789, 10.20179134868428084910907026044
