| 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
−15.16470292412512834171192263018, −13.47623301599466441924976443658, −12.58485271221844659635023497143, −11.73045273638713559260008104600, −9.552395157132788037876799218234, −8.884833520070222555446603737435, −7.978970910374215595369168709400, −6.59208060385067457332632319034, −5.04562045167097750618800522260, −2.71360369642394514686161012859,
2.17143491521171440875439069261, 3.95184861561075021569468048778, 6.22337091033520949129752970490, 7.43456585438114289135242459284, 9.120288601078986700257877850479, 10.13607100396976534364592373989, 10.74410280041591006542010995370, 11.91013611455763156873890476387, 13.69558300534269946655119819270, 14.31043290213389197967246072692
