| L(s) = 1 | + (−1.95 + 0.416i)2-s + (3.65 − 1.63i)4-s − 1.84·5-s + 7.45i·7-s + (−6.46 + 4.71i)8-s + (3.61 − 0.768i)10-s − 1.00i·11-s + 16.3·13-s + (−3.10 − 14.5i)14-s + (10.6 − 11.9i)16-s + 10.6·17-s + 4.35i·19-s + (−6.74 + 3.00i)20-s + (0.417 + 1.95i)22-s + 10.1i·23-s + ⋯ |
| L(s) = 1 | + (−0.978 + 0.208i)2-s + (0.913 − 0.407i)4-s − 0.369·5-s + 1.06i·7-s + (−0.808 + 0.588i)8-s + (0.361 − 0.0768i)10-s − 0.0910i·11-s + 1.25·13-s + (−0.221 − 1.04i)14-s + (0.667 − 0.744i)16-s + 0.624·17-s + 0.229i·19-s + (−0.337 + 0.150i)20-s + (0.0189 + 0.0890i)22-s + 0.441i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8430704294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8430704294\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.95 - 0.416i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 1.84T + 25T^{2} \) |
| 7 | \( 1 - 7.45iT - 49T^{2} \) |
| 11 | \( 1 + 1.00iT - 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 - 10.6T + 289T^{2} \) |
| 23 | \( 1 - 10.1iT - 529T^{2} \) |
| 29 | \( 1 + 22.3T + 841T^{2} \) |
| 31 | \( 1 + 27.3iT - 961T^{2} \) |
| 37 | \( 1 + 22.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 22.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 31.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 52.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 92.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 89.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 41.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 16.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 59.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 117.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 94.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 17.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 125.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 69.2T + 9.40e3T^{2} \) |
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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
−10.45182652706114627599019653763, −9.503315748758584818317002556831, −8.812176635962612464411452229958, −8.076668561799137412201714022218, −7.29420528316386986081109045587, −5.98136256907993017797099373026, −5.65069056076262076433143236022, −3.86308363676819235768720173073, −2.61709350957694909552714109036, −1.30386792387679387039691598306,
0.45599958969848779706567635494, 1.67592375021092923990025269573, 3.32456899975786929015011967310, 4.06770049023259466529698290977, 5.70228622981642099719473779637, 6.80514318006104021060663970052, 7.48063972369725768320313203883, 8.321797920948768960650147837089, 9.086370544548908376748462358810, 10.18092603887137683074563348186
