L(s) = 1 | + (−8.27 + 4.77i)5-s + (−8.03 − 4.63i)11-s + 4.24·13-s + (13.4 + 7.77i)17-s + (−17.4 − 30.2i)19-s + (−15.3 + 8.87i)23-s + (33.1 − 57.4i)25-s − 26.2i·29-s + (−16.0 + 27.7i)31-s + (−27.6 − 47.9i)37-s + 38.4i·41-s + 29.3·43-s + (−59.0 + 34.1i)47-s + (0.943 + 0.544i)53-s + 88.6·55-s + ⋯ |
L(s) = 1 | + (−1.65 + 0.955i)5-s + (−0.730 − 0.421i)11-s + 0.326·13-s + (0.792 + 0.457i)17-s + (−0.918 − 1.59i)19-s + (−0.668 + 0.386i)23-s + (1.32 − 2.29i)25-s − 0.904i·29-s + (−0.517 + 0.895i)31-s + (−0.747 − 1.29i)37-s + 0.937i·41-s + 0.682·43-s + (−1.25 + 0.725i)47-s + (0.0177 + 0.0102i)53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8891473295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8891473295\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (8.27 - 4.77i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (8.03 + 4.63i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 4.24T + 169T^{2} \) |
| 17 | \( 1 + (-13.4 - 7.77i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (17.4 + 30.2i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (15.3 - 8.87i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 26.2iT - 841T^{2} \) |
| 31 | \( 1 + (16.0 - 27.7i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (27.6 + 47.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 38.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 29.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + (59.0 - 34.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-0.943 - 0.544i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-59.0 - 34.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.8 - 41.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-4.32 + 7.49i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 95.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-45.0 + 77.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-74.3 - 128. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 88.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-66.0 + 38.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 12.7T + 9.40e3T^{2} \) |
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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
−8.994270894104003369391579494050, −8.159210332627851450647796811204, −7.70542592405318960870504736104, −6.90798136407626146340804357439, −6.13245525743693375688509816319, −4.96062307096023635875412873670, −3.99024118772659634814986701547, −3.33473954426364016843981195281, −2.41218325526112142560155263504, −0.53209870470383764361640049525,
0.47846291159754770010509120627, 1.80569821669473816115952609278, 3.38713064716182240469491278672, 3.98942891535241998638376206870, 4.87810390617329091593657754596, 5.61070032136658483997493378909, 6.85602310232794379041629774333, 7.74471771581139482406776591268, 8.173470690484352756714213159816, 8.763581454605319682621841861202