| L(s) = 1 | + (0.173 − 0.984i)2-s + (−1.44 + 2.05i)3-s + (−0.939 − 0.342i)4-s + (−0.949 + 2.02i)5-s + (1.77 + 1.77i)6-s + (1.60 + 0.140i)7-s + (−0.5 + 0.866i)8-s + (−1.13 − 3.10i)9-s + (1.82 + 1.28i)10-s + (−3.06 − 1.77i)11-s + (2.05 − 1.44i)12-s + (−4.50 − 1.63i)13-s + (0.415 − 1.55i)14-s + (−2.79 − 4.86i)15-s + (0.766 + 0.642i)16-s + (0.262 + 0.720i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.831 + 1.18i)3-s + (−0.469 − 0.171i)4-s + (−0.424 + 0.905i)5-s + (0.724 + 0.724i)6-s + (0.604 + 0.0529i)7-s + (−0.176 + 0.306i)8-s + (−0.376 − 1.03i)9-s + (0.578 + 0.406i)10-s + (−0.925 − 0.534i)11-s + (0.593 − 0.415i)12-s + (−1.24 − 0.454i)13-s + (0.111 − 0.414i)14-s + (−0.722 − 1.25i)15-s + (0.191 + 0.160i)16-s + (0.0636 + 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0357919 + 0.293346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0357919 + 0.293346i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (0.949 - 2.02i)T \) |
| 37 | \( 1 + (-5.79 + 1.84i)T \) |
| good | 3 | \( 1 + (1.44 - 2.05i)T + (-1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (-1.60 - 0.140i)T + (6.89 + 1.21i)T^{2} \) |
| 11 | \( 1 + (3.06 + 1.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.50 + 1.63i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.262 - 0.720i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (4.90 + 3.43i)T + (6.49 + 17.8i)T^{2} \) |
| 23 | \( 1 + (-3.85 - 6.67i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0899 - 0.0241i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (6.67 - 6.67i)T - 31iT^{2} \) |
| 41 | \( 1 + (-0.750 + 2.06i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 + (0.837 - 3.12i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-8.99 + 0.787i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-1.33 + 0.116i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-0.635 + 0.296i)T + (39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (0.828 - 9.47i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (1.49 + 8.49i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-4.69 - 4.69i)T + 73iT^{2} \) |
| 79 | \( 1 + (1.30 - 14.8i)T + (-77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (6.70 - 14.3i)T + (-53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (-1.27 - 14.5i)T + (-87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (2.01 - 1.16i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.30253005231223049659260466619, −11.01614064083519231468422730884, −10.33643685702499135404725183009, −9.508453133854821660199003865381, −8.220578292908512872299201438272, −7.08219250699649031461957333350, −5.52961453448742198163354105558, −4.93310793098970464337295789849, −3.78056266291391400993094769830, −2.61701912828955710804796926854,
0.20206860491280140684104755343, 2.01271622840188941689504797682, 4.46890719100768606195734912190, 5.12522347861405358313124490315, 6.20393579493746865439717246617, 7.31136024347769261317007857569, 7.79745735131571338548684538172, 8.754307822289853034841767719229, 10.05183629011888071402938541633, 11.31827737121394489201485484068
