| L(s) = 1 | − 1.55i·2-s + 1.73·3-s + 1.58·4-s − 2.68i·6-s + (−5.94 − 3.69i)7-s − 8.67i·8-s + 2.99·9-s − 16.2·11-s + 2.75·12-s − 20.3·13-s + (−5.73 + 9.23i)14-s − 7.11·16-s − 5.47·17-s − 4.65i·18-s − 4.44i·19-s + ⋯ |
| L(s) = 1 | − 0.776i·2-s + 0.577·3-s + 0.397·4-s − 0.448i·6-s + (−0.849 − 0.527i)7-s − 1.08i·8-s + 0.333·9-s − 1.47·11-s + 0.229·12-s − 1.56·13-s + (−0.409 + 0.659i)14-s − 0.444·16-s − 0.322·17-s − 0.258i·18-s − 0.233i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0921i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.147713477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147713477\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (5.94 + 3.69i)T \) |
| good | 2 | \( 1 + 1.55iT - 4T^{2} \) |
| 11 | \( 1 + 16.2T + 121T^{2} \) |
| 13 | \( 1 + 20.3T + 169T^{2} \) |
| 17 | \( 1 + 5.47T + 289T^{2} \) |
| 19 | \( 1 + 4.44iT - 361T^{2} \) |
| 23 | \( 1 + 16.0iT - 529T^{2} \) |
| 29 | \( 1 - 18.1T + 841T^{2} \) |
| 31 | \( 1 - 34.9iT - 961T^{2} \) |
| 37 | \( 1 + 0.793iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 61.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 75.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 66.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 35.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 89.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 92.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 101.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 8.19T + 6.88e3T^{2} \) |
| 89 | \( 1 - 68.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.0T + 9.40e3T^{2} \) |
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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
−10.32901110931239787328511241269, −9.653616796601709394269847596899, −8.514649270709656129082565840467, −7.24677160238514687699643466122, −6.94218085117813788413163156127, −5.35257957384433555293987186612, −4.08458595679685760849989583599, −2.87997698664952647397727898787, −2.31328356803508094748923820853, −0.36424861537198370538785266491,
2.34999536836933076095041807709, 2.92777707485577704113514602996, 4.74181283147969236798539789870, 5.68263476345785020033968381694, 6.63469525431740738326065038281, 7.64409457386697023996981253373, 8.048955935233557757615410181074, 9.343460173935757379198445015567, 9.985261182607981032274853427656, 11.02708221184950457746479117525
