| L(s) = 1 | + (−0.930 + 0.474i)2-s + (−2.78 − 0.440i)3-s + (−0.533 + 0.734i)4-s + (−1.77 + 1.35i)5-s + (2.79 − 0.909i)6-s + (0.0860 + 0.543i)7-s + (0.475 − 3.00i)8-s + (4.68 + 1.52i)9-s + (1.01 − 2.10i)10-s + (−3.29 − 0.335i)11-s + (1.80 − 1.80i)12-s + (1.47 + 2.89i)13-s + (−0.337 − 0.464i)14-s + (5.54 − 2.99i)15-s + (0.419 + 1.29i)16-s + (−2.57 + 5.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 + 0.335i)2-s + (−1.60 − 0.254i)3-s + (−0.266 + 0.367i)4-s + (−0.794 + 0.607i)5-s + (1.14 − 0.371i)6-s + (0.0325 + 0.205i)7-s + (0.168 − 1.06i)8-s + (1.56 + 0.507i)9-s + (0.319 − 0.666i)10-s + (−0.994 − 0.101i)11-s + (0.522 − 0.522i)12-s + (0.408 + 0.801i)13-s + (−0.0902 − 0.124i)14-s + (1.43 − 0.772i)15-s + (0.104 + 0.322i)16-s + (−0.623 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0285894 + 0.167267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0285894 + 0.167267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.77 - 1.35i)T \) |
| 11 | \( 1 + (3.29 + 0.335i)T \) |
| good | 2 | \( 1 + (0.930 - 0.474i)T + (1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (2.78 + 0.440i)T + (2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.0860 - 0.543i)T + (-6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (-1.47 - 2.89i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.57 - 5.04i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.25 - 0.914i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.803 + 0.803i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.44 + 2.50i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.509 - 1.56i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.945 - 0.149i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.25 - 7.23i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.55 - 2.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.636 + 4.02i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-6.27 + 3.19i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (3.97 - 5.47i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.75 - 2.84i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.62 - 2.62i)T - 67iT^{2} \) |
| 71 | \( 1 + (-2.11 - 6.51i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.96 - 1.57i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.28 - 3.96i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-10.0 - 5.14i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (7.56 + 14.8i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−16.20471740163066784494657306267, −15.21797524310911927942871349716, −13.23307867828626182446021915100, −12.25507812335793588808470534594, −11.19863289618888295913599438237, −10.30110651472893731285097129196, −8.447443235788767794372595750528, −7.23561140345853878943753894305, −6.13509483799179468359776873078, −4.25292693244045137084668134555,
0.37956536279088003648223298822, 4.68785534900789650180266309511, 5.60466936869180153713933009571, 7.56910618015951795584251758372, 9.107038245809173783026941094478, 10.53375170135071189292352337481, 11.08057689761349923192981339155, 12.18804476476780131690585776008, 13.40043347342704795004521847531, 15.35243268677487568655446903069
