| L(s) = 1 | + (1.39 − 0.245i)2-s + (1.87 − 0.684i)4-s + (1.49 − 1.77i)5-s + (5.91 + 2.15i)7-s + (2.44 − 1.41i)8-s + (1.64 − 2.84i)10-s + (−1.00 − 1.20i)11-s + (1.33 − 7.56i)13-s + (8.76 + 1.54i)14-s + (3.06 − 2.57i)16-s + (20.1 + 11.6i)17-s + (−15.0 − 26.1i)19-s + (1.58 − 4.36i)20-s + (−1.69 − 1.42i)22-s + (7.69 + 21.1i)23-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (0.469 − 0.171i)4-s + (0.298 − 0.355i)5-s + (0.844 + 0.307i)7-s + (0.306 − 0.176i)8-s + (0.164 − 0.284i)10-s + (−0.0916 − 0.109i)11-s + (0.102 − 0.581i)13-s + (0.625 + 0.110i)14-s + (0.191 − 0.160i)16-s + (1.18 + 0.684i)17-s + (−0.793 − 1.37i)19-s + (0.0794 − 0.218i)20-s + (−0.0772 − 0.0647i)22-s + (0.334 + 0.919i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.36480 - 0.444819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.36480 - 0.444819i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.245i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.49 + 1.77i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-5.91 - 2.15i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (1.00 + 1.20i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 7.56i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-20.1 - 11.6i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (15.0 + 26.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.69 - 21.1i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (49.0 - 8.65i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (27.8 - 10.1i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-14.5 + 25.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (17.1 + 3.02i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (64.1 - 53.7i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (11.3 - 31.1i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 86.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (29.2 - 34.9i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-79.4 - 28.9i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (8.80 - 49.9i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-32.5 - 18.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (3.93 + 6.82i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (12.5 + 71.1i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-25.4 + 4.47i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-23.4 + 13.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (27.0 - 22.6i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−12.89087661138680668888658565959, −11.53775666158304019327305382320, −10.90115097861319202630902369256, −9.573203754009178522747557277506, −8.395097043032783759716435044551, −7.25803406592059406062339801837, −5.70638318622645712217985522434, −5.01011828567882469000551623196, −3.42983471404357652342913986396, −1.68586186662809052143943098839,
1.95971867411037132248723291322, 3.70464878943411960322127401298, 4.97425807370810055361918333484, 6.14590695166224482876626458894, 7.33398320454638564975249391192, 8.332702628923155635682896480186, 9.871096173318722617103932197141, 10.82984701955113597280125686663, 11.78788367691477762870237750795, 12.71080534497298445830549117839
