| 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
−16.77964948538291144925352929126, −15.57187492851390711928685876983, −14.13584382873080753744966480447, −12.84329176083340508269056271789, −11.13627583905728267280884431525, −10.01956188479799134691794560608, −9.545669325874640927636910967801, −7.19649176680977479237125054795, −6.23787415338927850991396932418, −4.16104993419815656985370823560,
1.32958278132230922835767916557, 5.55529506519037425305069734249, 6.13863187399247579116405100545, 8.695940325029417786342541188184, 9.378176749327833674358759956902, 10.93868028522390048639669519041, 12.05093401046337307426188898549, 13.08852618232821704821466810866, 14.24756785921360593914637683710, 16.38928326642390933741707306336
