| L(s) = 1 | + (0.639 − 2.38i)2-s + (2.11 − 2.13i)3-s + (−1.82 − 1.05i)4-s + (−3.10 + 3.91i)5-s + (−3.73 − 6.40i)6-s + (−2.77 + 10.3i)7-s + (3.30 − 3.30i)8-s + (−0.0915 − 8.99i)9-s + (7.36 + 9.92i)10-s + (−5.66 − 9.81i)11-s + (−6.09 + 1.66i)12-s + (2.53 + 9.44i)13-s + (22.9 + 13.2i)14-s + (1.79 + 14.8i)15-s + (−9.99 − 17.3i)16-s + (8.05 + 8.05i)17-s + ⋯ |
| L(s) = 1 | + (0.319 − 1.19i)2-s + (0.703 − 0.710i)3-s + (−0.456 − 0.263i)4-s + (−0.621 + 0.783i)5-s + (−0.623 − 1.06i)6-s + (−0.396 + 1.48i)7-s + (0.413 − 0.413i)8-s + (−0.0101 − 0.999i)9-s + (0.736 + 0.992i)10-s + (−0.514 − 0.891i)11-s + (−0.508 + 0.138i)12-s + (0.194 + 0.726i)13-s + (1.63 + 0.946i)14-s + (0.119 + 0.992i)15-s + (−0.624 − 1.08i)16-s + (0.473 + 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0678 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0678 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04198 - 0.973519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04198 - 0.973519i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.11 + 2.13i)T \) |
| 5 | \( 1 + (3.10 - 3.91i)T \) |
| good | 2 | \( 1 + (-0.639 + 2.38i)T + (-3.46 - 2i)T^{2} \) |
| 7 | \( 1 + (2.77 - 10.3i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (5.66 + 9.81i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-2.53 - 9.44i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (-8.05 - 8.05i)T + 289iT^{2} \) |
| 19 | \( 1 - 3.73iT - 361T^{2} \) |
| 23 | \( 1 + (3.60 + 13.4i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (28.7 - 16.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-7.67 + 13.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-13.7 - 13.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-4.87 + 8.44i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-38.3 - 10.2i)T + (1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (0.451 - 1.68i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (45.3 - 45.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-10.2 - 5.91i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (28.0 + 48.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-47.4 + 12.7i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 22.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-37.2 + 37.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (-105. + 60.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-105. - 28.1i)T + (5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 + 16.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (46.1 - 172. i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−15.07510437184357621783587180817, −13.94012158715179244267511084656, −12.69776411778712435553919974527, −11.96927249754256703292659720711, −10.92903858573706346729463345487, −9.305486795745478406848939086135, −7.933109147849307879280457738863, −6.30059681113349294979115879236, −3.50460185932627643497495028602, −2.41479343500399823251895378311,
3.97682244398515980539783315653, 5.15549492303648857851414676766, 7.35170710858317958525026134980, 7.977529787888644353664953601169, 9.658448956958566744314515287648, 10.89681479703910610944680916340, 12.97843334698844180006114469587, 13.82717074971802764277485103672, 15.02252169090559841218809396795, 15.83708850750439460162088800824
