| L(s) = 1 | + (0.328 − 3.75i)2-s + (−10.0 − 1.77i)4-s + (1.58 − 4.74i)5-s + (−2.49 − 1.74i)7-s + (−6.04 + 22.5i)8-s + (−17.2 − 7.50i)10-s + (−8.21 − 2.98i)11-s + (−1.30 − 14.9i)13-s + (−7.38 + 8.80i)14-s + (44.3 + 16.1i)16-s + (6.46 + 24.1i)17-s + (−23.0 + 13.2i)19-s + (−24.3 + 44.8i)20-s + (−13.9 + 29.8i)22-s + (23.7 − 16.6i)23-s + ⋯ |
| L(s) = 1 | + (0.164 − 1.87i)2-s + (−2.51 − 0.442i)4-s + (0.317 − 0.948i)5-s + (−0.357 − 0.249i)7-s + (−0.755 + 2.81i)8-s + (−1.72 − 0.750i)10-s + (−0.746 − 0.271i)11-s + (−0.100 − 1.15i)13-s + (−0.527 + 0.629i)14-s + (2.77 + 1.00i)16-s + (0.380 + 1.42i)17-s + (−1.21 + 0.699i)19-s + (−1.21 + 2.24i)20-s + (−0.632 + 1.35i)22-s + (1.03 − 0.724i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.583113 + 0.359379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583113 + 0.359379i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.58 + 4.74i)T \) |
| good | 2 | \( 1 + (-0.328 + 3.75i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (2.49 + 1.74i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (8.21 + 2.98i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (1.30 + 14.9i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-6.46 - 24.1i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (23.0 - 13.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-23.7 + 16.6i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-3.19 - 3.80i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (2.35 - 13.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (5.80 + 21.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-10.7 - 9.04i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-10.1 - 21.6i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-19.7 - 13.8i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (35.7 + 35.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (7.15 + 19.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (13.4 + 76.0i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (102. - 8.95i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-12.6 + 21.9i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (23.2 - 86.6i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-6.63 - 7.91i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (69.0 + 6.03i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-61.3 + 35.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (69.9 - 32.6i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−10.46422073352666345960070966632, −9.786794902250222249253922508625, −8.655886733200764587130631192405, −8.133078378954680823745526354171, −5.95723578058612911198794365301, −5.02931972200633842830487774304, −3.99881869959451102663496994340, −2.90328263443952517471301900320, −1.58657313074583518522309295667, −0.28473664836349593034605039393,
2.81321580517075187758395337847, 4.35408385336706071479570547641, 5.34074148279054189669317301841, 6.33299626094710506258443167271, 7.05521688604168821624791883462, 7.63940311840586968243872583311, 9.023809208447250241504911972269, 9.472285831513793730107857676831, 10.66677123321775083588298535894, 11.93658363202045846056059463370
