| L(s) = 1 | + (−1.06 − 0.514i)2-s + (−1.90 − 1.29i)3-s + (−0.371 − 0.465i)4-s + (2.37 − 2.20i)5-s + (1.36 + 2.36i)6-s + (−1.38 + 2.39i)7-s + (0.684 + 2.99i)8-s + (0.840 + 2.14i)9-s + (−3.67 + 1.13i)10-s + (3.47 − 4.36i)11-s + (0.102 + 1.36i)12-s + (1.57 + 0.484i)13-s + (2.71 − 1.84i)14-s + (−7.38 + 1.11i)15-s + (0.546 − 2.39i)16-s + (0.555 + 0.515i)17-s + ⋯ |
| L(s) = 1 | + (−0.755 − 0.363i)2-s + (−1.09 − 0.748i)3-s + (−0.185 − 0.232i)4-s + (1.06 − 0.986i)5-s + (0.556 + 0.964i)6-s + (−0.523 + 0.906i)7-s + (0.241 + 1.06i)8-s + (0.280 + 0.713i)9-s + (−1.16 + 0.358i)10-s + (1.04 − 1.31i)11-s + (0.0295 + 0.394i)12-s + (0.435 + 0.134i)13-s + (0.724 − 0.493i)14-s + (−1.90 + 0.287i)15-s + (0.136 − 0.598i)16-s + (0.134 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.280563 - 0.346115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.280563 - 0.346115i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.54 - 0.330i)T \) |
| good | 2 | \( 1 + (1.06 + 0.514i)T + (1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (1.90 + 1.29i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-2.37 + 2.20i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (1.38 - 2.39i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.47 + 4.36i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 0.484i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-0.555 - 0.515i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.795 - 2.02i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (-0.00139 - 0.000210i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (3.87 - 2.63i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.301 - 4.02i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (0.999 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.30 - 3.03i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (4.90 + 6.14i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (3.76 - 1.16i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.811 - 3.55i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.639 - 8.52i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-3.43 + 8.73i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (-6.83 + 1.02i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (10.6 + 3.27i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (3.26 - 5.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.71 + 3.21i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (7.13 + 4.86i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (11.0 - 13.8i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−16.38928326642390933741707306336, −14.24756785921360593914637683710, −13.08852618232821704821466810866, −12.05093401046337307426188898549, −10.93868028522390048639669519041, −9.378176749327833674358759956902, −8.695940325029417786342541188184, −6.13863187399247579116405100545, −5.55529506519037425305069734249, −1.32958278132230922835767916557,
4.16104993419815656985370823560, 6.23787415338927850991396932418, 7.19649176680977479237125054795, 9.545669325874640927636910967801, 10.01956188479799134691794560608, 11.13627583905728267280884431525, 12.84329176083340508269056271789, 14.13584382873080753744966480447, 15.57187492851390711928685876983, 16.77964948538291144925352929126
