| L(s) = 1 | + (−1.34 − 0.426i)2-s + (0.438 − 1.67i)3-s + (1.63 + 1.15i)4-s + (1.15 − 0.351i)5-s + (−1.30 + 2.07i)6-s + (0.316 − 1.59i)7-s + (−1.71 − 2.25i)8-s + (−2.61 − 1.46i)9-s + (−1.71 − 0.0207i)10-s + (0.0667 − 0.677i)11-s + (2.64 − 2.23i)12-s + (−1.42 − 0.432i)13-s + (−1.10 + 2.00i)14-s + (−0.0808 − 2.09i)15-s + (1.34 + 3.76i)16-s + (1.97 − 0.819i)17-s + ⋯ |
| L(s) = 1 | + (−0.953 − 0.301i)2-s + (0.253 − 0.967i)3-s + (0.817 + 0.575i)4-s + (0.517 − 0.157i)5-s + (−0.533 + 0.845i)6-s + (0.119 − 0.601i)7-s + (−0.605 − 0.795i)8-s + (−0.871 − 0.489i)9-s + (−0.540 − 0.00655i)10-s + (0.0201 − 0.204i)11-s + (0.763 − 0.645i)12-s + (−0.395 − 0.119i)13-s + (−0.295 + 0.537i)14-s + (−0.0208 − 0.540i)15-s + (0.337 + 0.941i)16-s + (0.479 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.458183 - 0.854270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458183 - 0.854270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 + 0.426i)T \) |
| 3 | \( 1 + (-0.438 + 1.67i)T \) |
| good | 5 | \( 1 + (-1.15 + 0.351i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (-0.316 + 1.59i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.0667 + 0.677i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (1.42 + 0.432i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-1.97 + 0.819i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.76 + 1.48i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (-3.71 + 5.55i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.000620 + 0.00629i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (0.512 - 0.512i)T - 31iT^{2} \) |
| 37 | \( 1 + (8.88 + 4.74i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (4.32 - 6.46i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (3.35 + 4.08i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-4.53 + 1.87i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.129 - 1.31i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (-5.49 + 1.66i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-2.87 + 3.50i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-7.49 - 6.15i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (0.0838 - 0.421i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-8.22 + 1.63i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (2.88 - 6.96i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-9.81 + 5.24i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (10.6 - 7.09i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \) |
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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
−10.97452839886729583519817077550, −10.03213227916892426331236698223, −9.116761454365351287760673792844, −8.287569216190126842179445237036, −7.33102313250354870300501739459, −6.70932023848387654491159820883, −5.41442953093365972369138285927, −3.43329431346409709087093473469, −2.20491332020141823695180708851, −0.863810100009104387779432971783,
2.01173055148829531801514717513, 3.32179456304963941993208146750, 5.14107641148062972312558891965, 5.79201436116112105674759059718, 7.12550998672616697229300784902, 8.195427026487244573376114376971, 9.048346307065973597746863903498, 9.795718829106545818372751316446, 10.33166648075934203690825244879, 11.41467818578206958442506541554
