L(s) = 1 | + (−0.5 − 0.866i)5-s − 4·7-s + 3·11-s + (−3 + 5.19i)13-s + (1 + 1.73i)17-s + (3.5 − 2.59i)19-s + (2 − 3.46i)23-s + (−0.499 + 0.866i)25-s + (0.5 − 0.866i)29-s − 5·31-s + (2 + 3.46i)35-s − 4·37-s + (1 + 1.73i)41-s + (3 − 5.19i)47-s + 9·49-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s − 1.51·7-s + 0.904·11-s + (−0.832 + 1.44i)13-s + (0.242 + 0.420i)17-s + (0.802 − 0.596i)19-s + (0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s + (0.0928 − 0.160i)29-s − 0.898·31-s + (0.338 + 0.585i)35-s − 0.657·37-s + (0.156 + 0.270i)41-s + (0.437 − 0.757i)47-s + 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8689719260\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8689719260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-3.5 + 2.59i)T \) |
good | 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + (-8.5 + 14.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6 + 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.651901881445845856311804418725, −7.45990197158958493512580376178, −6.83641007431330357023185711493, −6.38226162138481808338497080014, −5.36841780246809574208281192990, −4.43717947030611179409087808746, −3.73002428415643947672550827682, −2.88007166650567570647748443265, −1.70956062175205331654767734095, −0.31317974222939298303837249091,
0.984050751173737153450723924121, 2.58118126271911375319498048089, 3.33588375185845154265011554385, 3.81358153883132156076803315015, 5.20924209378996784352518722786, 5.76357538241750172154811519651, 6.68645229352955628743311182619, 7.24937173782019793001168084143, 7.88049762224306117566402467395, 8.966891459658000588404092848485