| L(s) = 1 | + (−0.640 − 0.881i)2-s + (0.951 + 0.309i)3-s + (0.251 − 0.772i)4-s + (0.741 + 2.10i)5-s + (−0.336 − 1.03i)6-s − 3.08i·7-s + (−2.91 + 0.947i)8-s + (0.809 + 0.587i)9-s + (1.38 − 2.00i)10-s + (0.929 − 0.674i)11-s + (0.477 − 0.657i)12-s + (−2.39 + 3.30i)13-s + (−2.72 + 1.97i)14-s + (0.0527 + 2.23i)15-s + (1.38 + 1.00i)16-s + (−4.40 + 1.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.452 − 0.623i)2-s + (0.549 + 0.178i)3-s + (0.125 − 0.386i)4-s + (0.331 + 0.943i)5-s + (−0.137 − 0.423i)6-s − 1.16i·7-s + (−1.03 + 0.334i)8-s + (0.269 + 0.195i)9-s + (0.438 − 0.633i)10-s + (0.280 − 0.203i)11-s + (0.137 − 0.189i)12-s + (−0.665 + 0.915i)13-s + (−0.727 + 0.528i)14-s + (0.0136 + 0.577i)15-s + (0.346 + 0.252i)16-s + (−1.06 + 0.346i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.841107 - 0.340634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841107 - 0.340634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-0.741 - 2.10i)T \) |
| good | 2 | \( 1 + (0.640 + 0.881i)T + (-0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + 3.08iT - 7T^{2} \) |
| 11 | \( 1 + (-0.929 + 0.674i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.39 - 3.30i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (4.40 - 1.42i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.84 - 5.67i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.36 + 1.88i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.63 + 5.02i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.182 + 0.560i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.70 + 9.22i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (7.67 + 5.57i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.42iT - 43T^{2} \) |
| 47 | \( 1 + (-5.75 - 1.86i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.08 - 1.00i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.57 - 1.87i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (11.1 - 8.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.00 - 0.976i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.99 + 6.14i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.23 - 5.83i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.81 + 11.7i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-11.7 + 3.82i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.877 + 0.637i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.30 - 1.39i)T + (78.4 + 57.0i)T^{2} \) |
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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
−14.31043290213389197967246072692, −13.69558300534269946655119819270, −11.91013611455763156873890476387, −10.74410280041591006542010995370, −10.13607100396976534364592373989, −9.120288601078986700257877850479, −7.43456585438114289135242459284, −6.22337091033520949129752970490, −3.95184861561075021569468048778, −2.17143491521171440875439069261,
2.71360369642394514686161012859, 5.04562045167097750618800522260, 6.59208060385067457332632319034, 7.978970910374215595369168709400, 8.884833520070222555446603737435, 9.552395157132788037876799218234, 11.73045273638713559260008104600, 12.58485271221844659635023497143, 13.47623301599466441924976443658, 15.16470292412512834171192263018
