| L(s) = 1 | + (−0.412 − 1.35i)2-s + (2.71 + 1.24i)3-s + (−1.65 + 1.11i)4-s + (1.19 − 4.06i)5-s + (0.556 − 4.18i)6-s + (−1.26 + 3.65i)7-s + (2.19 + 1.78i)8-s + (3.88 + 4.48i)9-s + (−5.99 + 0.0637i)10-s + (1.50 − 0.364i)11-s + (−5.89 + 0.976i)12-s + (−0.635 − 1.23i)13-s + (5.46 + 0.202i)14-s + (8.29 − 9.56i)15-s + (1.50 − 3.70i)16-s + (3.35 + 1.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.291 − 0.956i)2-s + (1.56 + 0.716i)3-s + (−0.829 + 0.558i)4-s + (0.533 − 1.81i)5-s + (0.227 − 1.71i)6-s + (−0.478 + 1.38i)7-s + (0.776 + 0.630i)8-s + (1.29 + 1.49i)9-s + (−1.89 + 0.0201i)10-s + (0.452 − 0.109i)11-s + (−1.70 + 0.281i)12-s + (−0.176 − 0.341i)13-s + (1.46 + 0.0540i)14-s + (2.14 − 2.47i)15-s + (0.376 − 0.926i)16-s + (0.813 + 0.325i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89815 - 0.863099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89815 - 0.863099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.412 + 1.35i)T \) |
| 67 | \( 1 + (4.52 + 6.81i)T \) |
| good | 3 | \( 1 + (-2.71 - 1.24i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.19 + 4.06i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.26 - 3.65i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-1.50 + 0.364i)T + (9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (0.635 + 1.23i)T + (-7.54 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.35 - 1.34i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-6.07 + 2.10i)T + (14.9 - 11.7i)T^{2} \) |
| 23 | \( 1 + (2.12 + 2.98i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (3.82 - 2.20i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.798 - 0.411i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (1.39 + 0.806i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.00 - 4.71i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (9.85 - 1.41i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (-10.1 + 0.971i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (-3.02 - 0.435i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.505 - 0.786i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (5.52 + 1.34i)T + (54.2 + 27.9i)T^{2} \) |
| 71 | \( 1 + (10.3 - 4.14i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (0.257 - 1.06i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.340 - 7.14i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-4.53 - 4.75i)T + (-3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (2.18 + 4.78i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (6.11 - 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.16335937131169138612274623532, −9.577055461200226922729534690286, −9.048646352193771727531282984749, −8.593160276453122119824859323266, −7.85530082902277698165131225880, −5.55482357634174647340668094910, −4.82491736390370927900863637010, −3.64886217974906106713731465929, −2.67164214867608209135811789007, −1.53327584850138139674108388558,
1.62283459835472728637641441817, 3.22811769406187969628466742403, 3.80631471035782070396645407509, 5.89980773631308662851623127524, 6.99739987642340045850789938629, 7.23309730885308692426675693227, 7.81760323625707483383022048346, 9.211010542977138591069662000556, 9.928298344654989655682408286761, 10.32050901323339850956298596843
