| L(s) = 1 | + (1.82 − 0.616i)2-s + (−1.38 + 0.830i)3-s + (1.37 − 1.04i)4-s + (−0.455 − 1.14i)5-s + (−2.01 + 2.37i)6-s + (−0.696 − 0.322i)7-s + (−0.294 + 0.434i)8-s + (−0.188 + 0.355i)9-s + (−1.53 − 1.81i)10-s + (0.294 + 1.06i)11-s + (−1.03 + 2.58i)12-s + (−2.46 − 4.64i)13-s + (−1.47 − 0.160i)14-s + (1.57 + 1.19i)15-s + (−1.19 + 4.30i)16-s + (5.91 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (1.29 − 0.435i)2-s + (−0.797 + 0.479i)3-s + (0.687 − 0.522i)4-s + (−0.203 − 0.510i)5-s + (−0.822 + 0.968i)6-s + (−0.263 − 0.121i)7-s + (−0.104 + 0.153i)8-s + (−0.0628 + 0.118i)9-s + (−0.486 − 0.572i)10-s + (0.0889 + 0.320i)11-s + (−0.297 + 0.747i)12-s + (−0.682 − 1.28i)13-s + (−0.393 − 0.0428i)14-s + (0.407 + 0.309i)15-s + (−0.298 + 1.07i)16-s + (1.43 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16991 - 0.150059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16991 - 0.150059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (3.76 - 6.69i)T \) |
| good | 2 | \( 1 + (-1.82 + 0.616i)T + (1.59 - 1.21i)T^{2} \) |
| 3 | \( 1 + (1.38 - 0.830i)T + (1.40 - 2.65i)T^{2} \) |
| 5 | \( 1 + (0.455 + 1.14i)T + (-3.62 + 3.43i)T^{2} \) |
| 7 | \( 1 + (0.696 + 0.322i)T + (4.53 + 5.33i)T^{2} \) |
| 11 | \( 1 + (-0.294 - 1.06i)T + (-9.42 + 5.67i)T^{2} \) |
| 13 | \( 1 + (2.46 + 4.64i)T + (-7.29 + 10.7i)T^{2} \) |
| 17 | \( 1 + (-5.91 + 2.73i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (-3.52 - 0.776i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (-0.988 - 6.02i)T + (-21.7 + 7.34i)T^{2} \) |
| 29 | \( 1 + (2.10 + 0.708i)T + (23.0 + 17.5i)T^{2} \) |
| 31 | \( 1 + (9.66 - 2.12i)T + (28.1 - 13.0i)T^{2} \) |
| 37 | \( 1 + (0.500 + 0.737i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (0.770 - 4.70i)T + (-38.8 - 13.0i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 5.20i)T + (-36.8 - 22.1i)T^{2} \) |
| 47 | \( 1 + (1.55 - 3.90i)T + (-34.1 - 32.3i)T^{2} \) |
| 53 | \( 1 + (3.40 - 4.01i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (0.967 - 0.325i)T + (48.5 - 36.9i)T^{2} \) |
| 67 | \( 1 + (-7.01 + 10.3i)T + (-24.7 - 62.2i)T^{2} \) |
| 71 | \( 1 + (-0.493 + 1.23i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (-8.55 - 0.930i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-2.96 - 1.78i)T + (37.0 + 69.7i)T^{2} \) |
| 83 | \( 1 + (-0.0467 + 0.861i)T + (-82.5 - 8.97i)T^{2} \) |
| 89 | \( 1 + (-11.8 - 3.99i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (14.2 - 1.55i)T + (94.7 - 20.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.96078631671842394914891517220, −13.85682793379730356318618559028, −12.65925292531840403267503566799, −12.01118314154883648938153856291, −10.90538875704873381452101999355, −9.672275471878423306107604612844, −7.70612725693211622853553351072, −5.56020297840760984774448663695, −5.02271207368741185662724823212, −3.34339385042009392009645404912,
3.49745693430268563234986500635, 5.24798548844249535152396007978, 6.36680498226705385209679324568, 7.24432958307803920918367181445, 9.424870064751462360085307406287, 11.17138521413906780969100350072, 12.15505133129677747419728811088, 12.85092574480377442389507640360, 14.35879188014133130235470006086, 14.71741043106914607073806001866
