| L(s) = 1 | + 1.41·2-s + (−0.396 − 2.97i)3-s + 2.00·4-s + (−0.560 − 4.20i)6-s + 2.64i·7-s + 2.82·8-s + (−8.68 + 2.35i)9-s − 2.01i·11-s + (−0.792 − 5.94i)12-s + 11.6i·13-s + 3.74i·14-s + 4.00·16-s − 17.3·17-s + (−12.2 + 3.33i)18-s − 36.1·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.132 − 0.991i)3-s + 0.500·4-s + (−0.0934 − 0.700i)6-s + 0.377i·7-s + 0.353·8-s + (−0.965 + 0.261i)9-s − 0.183i·11-s + (−0.0660 − 0.495i)12-s + 0.892i·13-s + 0.267i·14-s + 0.250·16-s − 1.02·17-s + (−0.682 + 0.185i)18-s − 1.90·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4230970819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4230970819\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (0.396 + 2.97i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 2.01iT - 121T^{2} \) |
| 13 | \( 1 - 11.6iT - 169T^{2} \) |
| 17 | \( 1 + 17.3T + 289T^{2} \) |
| 19 | \( 1 + 36.1T + 361T^{2} \) |
| 23 | \( 1 + 32.2T + 529T^{2} \) |
| 29 | \( 1 - 46.1iT - 841T^{2} \) |
| 31 | \( 1 - 34.0T + 961T^{2} \) |
| 37 | \( 1 + 31.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 92.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 18.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 45.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 33.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 25.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 48.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 32.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 82.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 48.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 14.5iT - 9.40e3T^{2} \) |
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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
−10.22315393477463955468648732006, −8.789563406626434153690971081635, −8.439380171597442001108523698016, −7.19625882880125769538327162258, −6.53422921126782114451416618242, −5.94570391746343009582377149897, −4.83077873793687907598454917618, −3.85548033851784728765650737763, −2.44704490958066821809106947662, −1.79000420482747345122167003324,
0.091699467111568463842505575746, 2.19163351487226937944973359171, 3.28319220866068141519959206986, 4.40744705821052994688057958932, 4.66262591621437437297516917315, 6.13160399343286474118721864147, 6.38351591895990447972525706846, 7.973902177994478558342394142536, 8.449236741497325258519066822817, 9.849428675118928157673358502998
