| L(s) = 1 | + (−0.301 − 0.359i)2-s + (−2.39 − 2.00i)3-s + (0.309 − 1.75i)4-s + (−0.342 − 0.939i)5-s + 1.46i·6-s + (1.06 − 0.387i)7-s + (−1.53 + 0.886i)8-s + (1.17 + 6.64i)9-s + (−0.234 + 0.406i)10-s + (0.690 + 1.19i)11-s + (−4.25 + 3.57i)12-s + (−6.56 − 1.15i)13-s + (−0.460 − 0.265i)14-s + (−1.06 + 2.93i)15-s + (−2.56 − 0.933i)16-s + (3.42 − 0.603i)17-s + ⋯ |
| L(s) = 1 | + (−0.213 − 0.254i)2-s + (−1.38 − 1.15i)3-s + (0.154 − 0.876i)4-s + (−0.152 − 0.420i)5-s + 0.597i·6-s + (0.402 − 0.146i)7-s + (−0.542 + 0.313i)8-s + (0.390 + 2.21i)9-s + (−0.0741 + 0.128i)10-s + (0.208 + 0.360i)11-s + (−1.22 + 1.03i)12-s + (−1.82 − 0.321i)13-s + (−0.122 − 0.0710i)14-s + (−0.275 + 0.757i)15-s + (−0.640 − 0.233i)16-s + (0.830 − 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0534865 + 0.508942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0534865 + 0.508942i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.342 + 0.939i)T \) |
| 37 | \( 1 + (5.05 + 3.37i)T \) |
| good | 2 | \( 1 + (0.301 + 0.359i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (2.39 + 2.00i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.06 + 0.387i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.690 - 1.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.56 + 1.15i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.42 + 0.603i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-3.70 + 4.40i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.26 - 1.30i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.17 - 3.56i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.69iT - 31T^{2} \) |
| 41 | \( 1 + (-1.38 + 7.83i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 + (-0.933 + 1.61i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.92 - 0.700i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.08 + 5.74i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.18 - 1.09i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.97 + 2.17i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.10 + 2.60i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 0.109T + 73T^{2} \) |
| 79 | \( 1 + (0.920 + 2.52i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.40 - 13.6i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.36 + 12.0i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.43 - 1.98i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.00570812325497940269122550545, −11.30206255741144652181810086585, −10.35693615107249599427991622949, −9.297666845657597116148568266553, −7.49309244229707446966603132879, −7.00807808261679194356068656805, −5.38846329248678918496656453966, −5.14029662730295051024815635032, −2.01848948653591013571404861705, −0.57583149009773764933386539228,
3.30542626503241466669388524147, 4.55907178580488658674379876167, 5.66021692230609144916934351140, 6.87211764334778543456319781800, 7.931572078312834709852772919604, 9.462831431357054173496946420472, 10.12004609925302620478998703041, 11.42160781581958075219825274695, 11.79872986057226811448938962039, 12.64228626037768211948288785387
