| 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
−15.35243268677487568655446903069, −13.40043347342704795004521847531, −12.18804476476780131690585776008, −11.08057689761349923192981339155, −10.53375170135071189292352337481, −9.107038245809173783026941094478, −7.56910618015951795584251758372, −5.60466936869180153713933009571, −4.68785534900789650180266309511, −0.37956536279088003648223298822,
4.25292693244045137084668134555, 6.13509483799179468359776873078, 7.23561140345853878943753894305, 8.447443235788767794372595750528, 10.30110651472893731285097129196, 11.19863289618888295913599438237, 12.25507812335793588808470534594, 13.23307867828626182446021915100, 15.21797524310911927942871349716, 16.20471740163066784494657306267
