| L(s) = 1 | + (−1.72 − 0.107i)3-s + (0.902 + 0.159i)5-s + (−2.30 − 2.75i)7-s + (2.97 + 0.373i)9-s + (0.316 + 1.79i)11-s + (−4.18 + 1.52i)13-s + (−1.54 − 0.372i)15-s + (−3.92 − 2.26i)17-s + (0.794 − 0.458i)19-s + (3.69 + 5.00i)21-s + (−5.68 − 4.76i)23-s + (−3.90 − 1.42i)25-s + (−5.10 − 0.966i)27-s + (3.02 − 8.31i)29-s + (−3.84 + 4.58i)31-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0623i)3-s + (0.403 + 0.0711i)5-s + (−0.872 − 1.04i)7-s + (0.992 + 0.124i)9-s + (0.0952 + 0.540i)11-s + (−1.16 + 0.422i)13-s + (−0.398 − 0.0961i)15-s + (−0.951 − 0.549i)17-s + (0.182 − 0.105i)19-s + (0.806 + 1.09i)21-s + (−1.18 − 0.994i)23-s + (−0.781 − 0.284i)25-s + (−0.982 − 0.186i)27-s + (0.562 − 1.54i)29-s + (−0.690 + 0.823i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0573083 - 0.286668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0573083 - 0.286668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.107i)T \) |
| good | 5 | \( 1 + (-0.902 - 0.159i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.30 + 2.75i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.316 - 1.79i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (4.18 - 1.52i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.92 + 2.26i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.794 + 0.458i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.68 + 4.76i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.02 + 8.31i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.84 - 4.58i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.16 + 3.19i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.919 + 0.162i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.999 + 0.838i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 2.53iT - 53T^{2} \) |
| 59 | \( 1 + (2.13 - 12.1i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.66 - 2.23i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.986 - 2.71i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.44 + 7.69i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.528 + 0.915i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.99 + 10.9i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-15.7 - 5.75i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-13.1 + 7.58i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.86 - 10.5i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50570174491983190428722557333, −10.09225068920028080757248456621, −9.320211788591438464122236252991, −7.65078245810159157954660178853, −6.83037859985882421104235864830, −6.24326024384918833108205697658, −4.86135396822546849634655453339, −4.06838227844999666977493689746, −2.20347978803475183306369973065, −0.19422069732023627023845259700,
2.08514301770062651538403632174, 3.64096125538960380034815375561, 5.15130220625929019756501125912, 5.83235890474672530546992314082, 6.59622463501333486159535976086, 7.77608739955137019585122561589, 9.165639520573270459740667667307, 9.713502985616633889840702051580, 10.66655024324614660940004857023, 11.60564303365642797528493992743
