| L(s) = 1 | + (0.0735 + 0.0647i)2-s + (1.41 + 0.996i)3-s + (−0.250 − 1.97i)4-s + (3.04 + 0.442i)5-s + (0.0396 + 0.165i)6-s + (0.277 + 3.06i)7-s + (0.219 − 0.323i)8-s + (1.01 + 2.82i)9-s + (0.195 + 0.229i)10-s + (−0.670 + 0.682i)11-s + (1.61 − 3.04i)12-s + (−2.30 − 0.0834i)13-s + (−0.178 + 0.243i)14-s + (3.87 + 3.66i)15-s + (−3.81 + 0.985i)16-s + (1.60 − 0.743i)17-s + ⋯ |
| L(s) = 1 | + (0.0520 + 0.0458i)2-s + (0.817 + 0.575i)3-s + (−0.125 − 0.987i)4-s + (1.36 + 0.197i)5-s + (0.0161 + 0.0673i)6-s + (0.104 + 1.15i)7-s + (0.0776 − 0.114i)8-s + (0.337 + 0.941i)9-s + (0.0617 + 0.0726i)10-s + (−0.202 + 0.205i)11-s + (0.465 − 0.879i)12-s + (−0.640 − 0.0231i)13-s + (−0.0475 + 0.0650i)14-s + (0.999 + 0.945i)15-s + (−0.954 + 0.246i)16-s + (0.389 − 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.19600 + 0.588582i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19600 + 0.588582i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.41 - 0.996i)T \) |
| 59 | \( 1 + (7.23 - 2.57i)T \) |
| good | 2 | \( 1 + (-0.0735 - 0.0647i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (-3.04 - 0.442i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (-0.277 - 3.06i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (0.670 - 0.682i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.30 + 0.0834i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-1.60 + 0.743i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (-4.06 - 0.895i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (6.11 + 2.30i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (0.843 + 4.18i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (-2.10 + 6.65i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (3.60 + 5.31i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (0.383 + 0.313i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-8.43 + 2.17i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (-0.662 + 0.0963i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (-1.88 + 2.22i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (-0.721 - 0.635i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (-3.66 - 7.55i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (5.08 - 12.7i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (3.36 + 0.365i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-0.156 + 8.64i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (0.877 - 0.445i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (2.32 + 0.783i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (7.15 + 9.77i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−10.47174271338610775087047622550, −9.780578124614157428948730863959, −9.515773367377668268201339767541, −8.553085211246208178751869115052, −7.33720141305678654285189934167, −5.78036229555631072559875828687, −5.62836306385511755211412034281, −4.38944909409859720594415249958, −2.58155003680114977278282727555, −1.98471812961018527969600431113,
1.47416058479810308540031542936, 2.76563728464284044813292131062, 3.75958606081253900631006111542, 5.04838406869896112974298103000, 6.41502838046855150309656560330, 7.36754555768439978697679393167, 7.939137309473230605099009857752, 9.021040414242680163159360320635, 9.726467514085038654676820147282, 10.56119120991216229133777304792
