| L(s) = 1 | + (−0.903 − 1.24i)2-s + (−0.951 − 0.309i)3-s + (−0.112 + 0.346i)4-s + (−1.93 − 1.12i)5-s + (0.475 + 1.46i)6-s − 1.68i·7-s + (−2.39 + 0.777i)8-s + (0.809 + 0.587i)9-s + (0.349 + 3.42i)10-s + (2.40 − 1.75i)11-s + (0.214 − 0.294i)12-s + (0.136 − 0.188i)13-s + (−2.09 + 1.52i)14-s + (1.49 + 1.66i)15-s + (3.71 + 2.70i)16-s + (7.09 − 2.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.639 − 0.879i)2-s + (−0.549 − 0.178i)3-s + (−0.0563 + 0.173i)4-s + (−0.864 − 0.502i)5-s + (0.193 + 0.597i)6-s − 0.637i·7-s + (−0.845 + 0.274i)8-s + (0.269 + 0.195i)9-s + (0.110 + 1.08i)10-s + (0.726 − 0.527i)11-s + (0.0618 − 0.0851i)12-s + (0.0379 − 0.0522i)13-s + (−0.560 + 0.407i)14-s + (0.385 + 0.430i)15-s + (0.929 + 0.675i)16-s + (1.72 − 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.160080 - 0.480070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.160080 - 0.480070i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.93 + 1.12i)T \) |
| good | 2 | \( 1 + (0.903 + 1.24i)T + (-0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + 1.68iT - 7T^{2} \) |
| 11 | \( 1 + (-2.40 + 1.75i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.136 + 0.188i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-7.09 + 2.30i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.232 + 0.716i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.512 + 0.706i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.12 - 6.53i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.03 + 9.33i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.95 - 8.19i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.07 + 2.23i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.27iT - 43T^{2} \) |
| 47 | \( 1 + (-8.14 - 2.64i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.68 - 1.84i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.11 + 2.26i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.55 + 2.58i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.70 + 0.554i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.35 - 4.16i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.84 - 12.1i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.27 - 7.01i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.13 - 1.34i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.79 + 7.11i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (9.01 + 2.92i)T + (78.4 + 57.0i)T^{2} \) |
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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
−14.02333802345674893055456941635, −12.49468604115168415697600307258, −11.75935544510422939712117984486, −10.93897159702145443978758495717, −9.823443630242352308496149814365, −8.585486588160573683326669487492, −7.23715407818483127657603251238, −5.51561753168645456505214368323, −3.64007552872764747341719041804, −0.957459619413945795699613286555,
3.65203290699426218399510222371, 5.70606843969888096506646729875, 6.92792828023009224459658603150, 7.911605023279146021538817931817, 9.136008514816409944128365962871, 10.41657441348033795866753423639, 11.96123411884263762284321045102, 12.28158362724071700693361403484, 14.50291736657123728063095542117, 15.19640497143500829651723826846
