L(s) = 1 | + (−0.748 − 0.663i)2-s + (−0.970 + 0.239i)3-s + (0.120 + 0.992i)4-s + (0.885 − 0.464i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (0.568 − 0.822i)8-s + (0.885 − 0.464i)9-s + (−0.970 − 0.239i)10-s + (−0.354 − 0.935i)11-s + (−0.354 − 0.935i)12-s + (0.120 − 0.992i)13-s + (0.120 + 0.992i)14-s + (−0.748 + 0.663i)15-s + (−0.970 + 0.239i)16-s + (0.568 + 0.822i)17-s + ⋯ |
L(s) = 1 | + (−0.748 − 0.663i)2-s + (−0.970 + 0.239i)3-s + (0.120 + 0.992i)4-s + (0.885 − 0.464i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (0.568 − 0.822i)8-s + (0.885 − 0.464i)9-s + (−0.970 − 0.239i)10-s + (−0.354 − 0.935i)11-s + (−0.354 − 0.935i)12-s + (0.120 − 0.992i)13-s + (0.120 + 0.992i)14-s + (−0.748 + 0.663i)15-s + (−0.970 + 0.239i)16-s + (0.568 + 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0198 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0198 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3424877843 - 0.3357424895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3424877843 - 0.3357424895i\) |
\(L(1)\) |
\(\approx\) |
\(0.5321343173 - 0.2590884114i\) |
\(L(1)\) |
\(\approx\) |
\(0.5321343173 - 0.2590884114i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (-0.748 - 0.663i)T \) |
| 3 | \( 1 + (-0.970 + 0.239i)T \) |
| 5 | \( 1 + (0.885 - 0.464i)T \) |
| 7 | \( 1 + (-0.748 - 0.663i)T \) |
| 11 | \( 1 + (-0.354 - 0.935i)T \) |
| 13 | \( 1 + (0.120 - 0.992i)T \) |
| 17 | \( 1 + (0.568 + 0.822i)T \) |
| 19 | \( 1 + (0.120 - 0.992i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.354 + 0.935i)T \) |
| 31 | \( 1 + (-0.354 + 0.935i)T \) |
| 37 | \( 1 + (-0.970 + 0.239i)T \) |
| 41 | \( 1 + (-0.354 - 0.935i)T \) |
| 43 | \( 1 + (-0.970 - 0.239i)T \) |
| 47 | \( 1 + (0.885 + 0.464i)T \) |
| 59 | \( 1 + (0.885 + 0.464i)T \) |
| 61 | \( 1 + (0.568 - 0.822i)T \) |
| 67 | \( 1 + (0.120 + 0.992i)T \) |
| 71 | \( 1 + (-0.970 - 0.239i)T \) |
| 73 | \( 1 + (0.568 + 0.822i)T \) |
| 79 | \( 1 + (-0.748 + 0.663i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.568 + 0.822i)T \) |
| 97 | \( 1 + (0.885 - 0.464i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−33.63653392228576428410457671312, −33.21517850730752107046545845792, −31.56265812826811872184449327221, −29.71350161177565455356170232231, −28.84166023274688506268830013847, −28.18959263650212335423838909959, −26.72909692290306220752907240828, −25.51887141735663028927245196427, −24.78744947082757948827579357058, −23.2431411018332705106758205792, −22.4860578960457206067325154375, −20.97635478765946152534987153602, −18.86796074542560018627741131963, −18.42162790634526835286821958221, −17.17421376683755996894774779983, −16.25419118492275221723013449483, −14.914272092351730503658025333235, −13.34563867407908958080626976786, −11.74270148953157072830785110419, −10.19787324118481789988163780681, −9.4126926796884784551721532653, −7.27598264495185618344386428422, −6.28920722496971948110984756166, −5.22957463881822154513079225271, −1.96871538591134264192016974721,
0.97746821334734765209104852133, 3.36204948659211704297753310604, 5.36210668094186783033092246027, 6.90382504700526516140081716810, 8.827293008149977338440979990240, 10.19234373902293297497398409098, 10.827353451752792306538451916761, 12.57046561809275041616870347469, 13.32766903011590739762540562570, 15.919698453633477940197285444362, 16.89528733251139442914043628834, 17.62700366054847927063015306288, 18.9315273523695577925989373120, 20.36287849153590311400708600391, 21.467684909945464116956206091716, 22.324170530821241466598758152757, 23.80821321931470503989917842514, 25.36329069258860242535166297818, 26.47738311554575424015530388813, 27.62428406831909709384136436779, 28.69987188637676019210713826759, 29.376487711791626276607628274333, 30.169610320196112711654389640973, 32.23197792765326368432791508397, 33.04342158238739243186610907298