| L(s) = 1 | + (0.432 + 0.151i)3-s + (−1.40 − 0.320i)5-s + (−0.759 − 1.57i)7-s + (−2.18 − 1.73i)9-s + (0.426 − 3.78i)11-s + (−3.71 + 2.96i)13-s + (−0.558 − 0.350i)15-s + (1.08 − 1.08i)17-s + (−1.57 − 4.48i)19-s + (−0.0898 − 0.797i)21-s + (−0.0809 + 0.0184i)23-s + (−2.64 − 1.27i)25-s + (−1.41 − 2.24i)27-s + (3.03 + 4.44i)29-s + (8.91 − 5.60i)31-s + ⋯ |
| L(s) = 1 | + (0.249 + 0.0873i)3-s + (−0.627 − 0.143i)5-s + (−0.287 − 0.596i)7-s + (−0.727 − 0.579i)9-s + (0.128 − 1.14i)11-s + (−1.03 + 0.821i)13-s + (−0.144 − 0.0905i)15-s + (0.263 − 0.263i)17-s + (−0.360 − 1.03i)19-s + (−0.0196 − 0.174i)21-s + (−0.0168 + 0.00385i)23-s + (−0.528 − 0.254i)25-s + (−0.271 − 0.432i)27-s + (0.563 + 0.826i)29-s + (1.60 − 1.00i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.404465 - 0.687161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.404465 - 0.687161i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + (-3.03 - 4.44i)T \) |
| good | 3 | \( 1 + (-0.432 - 0.151i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (1.40 + 0.320i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (0.759 + 1.57i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.426 + 3.78i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (3.71 - 2.96i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.08 + 1.08i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.57 + 4.48i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (0.0809 - 0.0184i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-8.91 + 5.60i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (5.41 - 0.610i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (0.943 + 0.943i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.824 + 1.31i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (6.23 + 0.702i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (1.20 - 5.27i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 13.1iT - 59T^{2} \) |
| 61 | \( 1 + (0.0136 - 0.0389i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (-5.77 + 7.24i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (2.67 + 3.35i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-7.63 + 12.1i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (-5.46 + 0.616i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (5.03 - 10.4i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (2.84 + 4.52i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (-3.29 - 9.40i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−10.83425052977644722193873730974, −9.752449636299469113047579045908, −8.905853578243021280555464406093, −8.135154281934619845829127970425, −7.05042101926919263954610604592, −6.19449877650818291392112407579, −4.81573573652917167622783583542, −3.77248655630894694113549958077, −2.73864005154981613113038842211, −0.45548456559717816370641086771,
2.17953417817838588019758023822, 3.26194855317310529326761037321, 4.64210740826495191354345563455, 5.63085636206256354259704971809, 6.83294890190938577140423049001, 7.922522379946927416530586315594, 8.341252884944656620959997678714, 9.726157381873214768166074904786, 10.26050156583128793508453555152, 11.49791357624566246560267796985
