| L(s) = 1 | + (−0.161 − 1.01i)2-s + (−1.38 − 1.03i)3-s + (0.888 − 0.288i)4-s + (−2.26 − 1.15i)5-s + (−0.835 + 1.58i)6-s + (−1.08 + 0.260i)7-s + (−1.37 − 2.69i)8-s + (0.840 + 2.87i)9-s + (−0.812 + 2.49i)10-s + (1.53 − 1.80i)11-s + (−1.53 − 0.523i)12-s + (−0.726 − 1.18i)13-s + (0.441 + 1.06i)14-s + (1.94 + 3.95i)15-s + (−1.01 + 0.739i)16-s + (−1.52 + 0.120i)17-s + ⋯ |
| L(s) = 1 | + (−0.114 − 0.720i)2-s + (−0.800 − 0.599i)3-s + (0.444 − 0.144i)4-s + (−1.01 − 0.516i)5-s + (−0.341 + 0.645i)6-s + (−0.410 + 0.0985i)7-s + (−0.486 − 0.954i)8-s + (0.280 + 0.959i)9-s + (−0.256 + 0.790i)10-s + (0.463 − 0.542i)11-s + (−0.442 − 0.151i)12-s + (−0.201 − 0.328i)13-s + (0.117 + 0.284i)14-s + (0.501 + 1.02i)15-s + (−0.254 + 0.184i)16-s + (−0.370 + 0.0291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.193784 - 0.678824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193784 - 0.678824i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.38 + 1.03i)T \) |
| 41 | \( 1 + (4.94 - 4.06i)T \) |
| good | 2 | \( 1 + (0.161 + 1.01i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (2.26 + 1.15i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (1.08 - 0.260i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (-1.53 + 1.80i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.726 + 1.18i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (1.52 - 0.120i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-3.34 + 5.45i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-7.14 - 5.19i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.38 - 0.502i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (-5.33 - 1.73i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.788 + 2.42i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (6.72 - 1.06i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.86 + 7.77i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (0.400 - 5.09i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (3.09 - 4.25i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.173 - 1.09i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-7.65 - 8.96i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-0.729 - 0.622i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (0.605 - 0.605i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.43 + 13.1i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 0.110iT - 83T^{2} \) |
| 89 | \( 1 + (2.24 + 9.35i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (3.51 - 2.99i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.67781534985522021483635315549, −11.66325529242954424913498898300, −11.44022326284542478048124044837, −10.19481545958197190426324024993, −8.809635422953655085445608216795, −7.36873568838299986039425458035, −6.45943884889332523732959648445, −4.92812077307516048043267981955, −3.10934761837970152608682311137, −0.901520889645331297023265134035,
3.35899635055738277044906826862, 4.82504676655302103944333979580, 6.43977800720390475675973664378, 7.00559208016384193115975812117, 8.298771516393034151341671010366, 9.759282069802874028293341525467, 10.89701986858107832178774563297, 11.76224483676506277508164720436, 12.39097287228350909335801319831, 14.31913819366199505996975558903
