| L(s) = 1 | + (−4.67 − 1.77i)5-s − 9.67i·7-s + 8.58i·11-s − 19.0i·13-s − 26.2·17-s − 26.6·19-s − 25.8·23-s + (18.7 + 16.5i)25-s + 18.3i·29-s + 7.56·31-s + (−17.1 + 45.2i)35-s + 47.0i·37-s − 0.602i·41-s − 62.2i·43-s + 79.2·47-s + ⋯ |
| L(s) = 1 | + (−0.934 − 0.354i)5-s − 1.38i·7-s + 0.780i·11-s − 1.46i·13-s − 1.54·17-s − 1.40·19-s − 1.12·23-s + (0.748 + 0.663i)25-s + 0.633i·29-s + 0.243·31-s + (−0.490 + 1.29i)35-s + 1.27i·37-s − 0.0146i·41-s − 1.44i·43-s + 1.68·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3693540965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3693540965\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.67 + 1.77i)T \) |
| good | 7 | \( 1 + 9.67iT - 49T^{2} \) |
| 11 | \( 1 - 8.58iT - 121T^{2} \) |
| 13 | \( 1 + 19.0iT - 169T^{2} \) |
| 17 | \( 1 + 26.2T + 289T^{2} \) |
| 19 | \( 1 + 26.6T + 361T^{2} \) |
| 23 | \( 1 + 25.8T + 529T^{2} \) |
| 29 | \( 1 - 18.3iT - 841T^{2} \) |
| 31 | \( 1 - 7.56T + 961T^{2} \) |
| 37 | \( 1 - 47.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 0.602iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 79.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 15.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 62.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 48.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 99.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 60.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 77.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 28.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 23.3iT - 9.40e3T^{2} \) |
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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
−8.767481431806117153547027984923, −8.308186673216523821059431759955, −7.38112493236926714436762020224, −7.01265634261423549356667470916, −5.94452342456817243759195605395, −4.67507946075053921918137043778, −4.28625720462524095977880652932, −3.46327291747985199015098707775, −2.15377653019875470319429867648, −0.75190572412762293496804776055,
0.12566035570467189255189909390, 2.06788284377989074428568083138, 2.64761915605545553396169740500, 4.08108478622502857701766072504, 4.37565980028403929525772135351, 5.80027520943528213257954351681, 6.34399751351649491437923738059, 7.11485056684166560939632309963, 8.183976217269618576808322692661, 8.776958457061252613987336435474
