| 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
−15.00711274087839721086635475114, −14.20072354800474057058469972590, −12.99792453501419996691351705711, −11.68566659418431360059958624556, −10.09573122338856883773467233282, −9.052588461564207932062475332589, −7.970262499693222739094157899222, −7.05763039217707667658460593854, −4.26465972401635381716582849322, −2.50200019166499647861554253238,
3.57896798502844904725783157680, 4.26471297694770503060328064825, 7.54661833342103225805785412531, 8.435333080002113486741986474258, 8.810533700989884453592307985270, 10.25893358207880007337179742216, 12.09485476476113527845425693686, 13.24645045558262749451106063057, 14.18445284457449645194236190148, 15.36592613602952025711140177299
