| L(s) = 1 | + (−0.554 + 0.256i)2-s + (2.89 + 0.974i)3-s + (−1.05 + 1.23i)4-s + (−2.81 − 1.69i)5-s + (−1.85 + 0.201i)6-s + (0.0467 − 0.862i)7-s + (0.593 − 2.13i)8-s + (5.03 + 3.82i)9-s + (1.99 + 0.217i)10-s + (0.631 − 3.84i)11-s + (−4.25 + 2.55i)12-s + (−1.31 + 0.998i)13-s + (0.195 + 0.490i)14-s + (−6.50 − 7.65i)15-s + (−0.306 − 1.87i)16-s + (0.182 + 3.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.392 + 0.181i)2-s + (1.67 + 0.562i)3-s + (−0.526 + 0.619i)4-s + (−1.26 − 0.758i)5-s + (−0.757 + 0.0823i)6-s + (0.0176 − 0.325i)7-s + (0.209 − 0.755i)8-s + (1.67 + 1.27i)9-s + (0.632 + 0.0687i)10-s + (0.190 − 1.16i)11-s + (−1.22 + 0.738i)12-s + (−0.364 + 0.276i)13-s + (0.0522 + 0.131i)14-s + (−1.67 − 1.97i)15-s + (−0.0767 − 0.468i)16-s + (0.0442 + 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.852741 + 0.263658i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.852741 + 0.263658i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (3.82 - 6.65i)T \) |
| good | 2 | \( 1 + (0.554 - 0.256i)T + (1.29 - 1.52i)T^{2} \) |
| 3 | \( 1 + (-2.89 - 0.974i)T + (2.38 + 1.81i)T^{2} \) |
| 5 | \( 1 + (2.81 + 1.69i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.0467 + 0.862i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (-0.631 + 3.84i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (1.31 - 0.998i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (-0.182 - 3.36i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (3.16 - 2.99i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (2.04 - 0.450i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (3.93 + 1.82i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (-5.88 - 5.57i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (0.331 + 1.19i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (3.25 + 0.716i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (0.471 + 2.87i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (-5.86 + 3.52i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (-2.19 + 0.239i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 4.92i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (-1.13 + 4.10i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (9.08 - 5.46i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (4.53 + 11.3i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (9.88 - 3.33i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (-3.22 + 4.76i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (-2.24 - 1.03i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-3.53 + 8.87i)T + (-70.4 - 66.7i)T^{2} \) |
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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
−15.36592613602952025711140177299, −14.18445284457449645194236190148, −13.24645045558262749451106063057, −12.09485476476113527845425693686, −10.25893358207880007337179742216, −8.810533700989884453592307985270, −8.435333080002113486741986474258, −7.54661833342103225805785412531, −4.26471297694770503060328064825, −3.57896798502844904725783157680,
2.50200019166499647861554253238, 4.26465972401635381716582849322, 7.05763039217707667658460593854, 7.970262499693222739094157899222, 9.052588461564207932062475332589, 10.09573122338856883773467233282, 11.68566659418431360059958624556, 12.99792453501419996691351705711, 14.20072354800474057058469972590, 15.00711274087839721086635475114
