L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (2.32 − 4.02i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (3.81 − 2.20i)11-s + 0.999i·12-s + (−1.32 − 3.35i)13-s − 4.64·14-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s − 0.999·18-s + (6.96 + 4.02i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (0.877 − 1.52i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (1.14 − 0.663i)11-s + 0.288i·12-s + (−0.366 − 0.930i)13-s − 1.24·14-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s − 0.235·18-s + (1.59 + 0.922i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.960092386\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960092386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1.32 + 3.35i)T \) |
good | 7 | \( 1 + (-2.32 + 4.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.81 + 2.20i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.96 - 4.02i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.845 + 0.488i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.15 + 3.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.44iT - 31T^{2} \) |
| 37 | \( 1 + (1.89 + 3.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.31 - 3.64i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.620 - 0.358i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.75T + 47T^{2} \) |
| 53 | \( 1 - 13.5iT - 53T^{2} \) |
| 59 | \( 1 + (-1.88 - 1.09i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.73 - 6.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.912 - 1.58i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.88 + 3.97i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 + (7.07 - 4.08i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.91 + 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.925361683595156795880199570704, −8.111273555767693370280320491493, −7.47315579358811576027789165045, −6.92970550245167935241806747248, −5.62280257492731550090420616312, −4.50821273554505616838431891213, −3.75383324249149748895611571784, −2.96373516516642412627754848148, −1.52480289663271850324816430095, −0.828398917652786137410891022377,
1.63049433138051925642494887692, 2.38137917863710547163422267530, 3.81707990571778229703572103345, 4.82891607189675155307468820402, 5.33455072819961641140151118420, 6.48610450771706619912967612680, 7.14006830146499638248168439762, 8.033327350967850838644993533132, 8.918746702928164533026740498605, 9.167165598141037731550451483789