| L(s) = 1 | + (1.29 − 0.625i)2-s + (−1.72 + 0.149i)3-s + (0.0485 − 0.0608i)4-s + (−2.51 − 2.33i)5-s + (−2.14 + 1.27i)6-s + (0.896 + 2.48i)7-s + (−0.616 + 2.70i)8-s + (2.95 − 0.515i)9-s + (−4.72 − 1.45i)10-s + (2.86 + 1.95i)11-s + (−0.0746 + 0.112i)12-s + (3.25 + 2.21i)13-s + (2.72 + 2.67i)14-s + (4.68 + 3.64i)15-s + (0.923 + 4.04i)16-s + (−2.59 + 0.391i)17-s + ⋯ |
| L(s) = 1 | + (0.918 − 0.442i)2-s + (−0.996 + 0.0862i)3-s + (0.0242 − 0.0304i)4-s + (−1.12 − 1.04i)5-s + (−0.876 + 0.519i)6-s + (0.338 + 0.940i)7-s + (−0.217 + 0.955i)8-s + (0.985 − 0.171i)9-s + (−1.49 − 0.460i)10-s + (0.862 + 0.588i)11-s + (−0.0215 + 0.0323i)12-s + (0.902 + 0.615i)13-s + (0.727 + 0.714i)14-s + (1.20 + 0.941i)15-s + (0.230 + 1.01i)16-s + (−0.630 + 0.0950i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09523 + 0.495683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09523 + 0.495683i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 - 0.149i)T \) |
| 7 | \( 1 + (-0.896 - 2.48i)T \) |
| good | 2 | \( 1 + (-1.29 + 0.625i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (2.51 + 2.33i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-2.86 - 1.95i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-3.25 - 2.21i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (2.59 - 0.391i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.30 - 3.99i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.403 + 1.02i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (3.59 - 0.542i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 + (0.631 - 1.60i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (3.29 - 1.01i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (-10.2 - 3.15i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-4.61 + 2.22i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (0.613 + 1.56i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-2.97 - 13.0i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (2.00 + 2.51i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 1.42T + 67T^{2} \) |
| 71 | \( 1 + (5.28 - 6.62i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-13.5 + 9.23i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + (-13.2 + 9.04i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (0.312 + 4.17i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (6.82 - 11.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.60090750269382949133414638781, −10.95639930640416233643610172617, −9.278940796239506481849682872983, −8.649869321524629623754461001344, −7.56575294746097839153664912300, −6.14168089665337777800161449300, −5.24153917125559302168875385571, −4.34509180484814731934738985442, −3.80915940279879575389468187730, −1.65426088975420531367870868062,
0.70549669352861379554312781107, 3.60193691633532909002231519036, 4.06255184470114850114373960441, 5.26778866858226273876336658494, 6.34870512346136221462771424723, 6.99221661031242577290521465415, 7.71404140957434722408918877107, 9.290689355167905841005241994652, 10.67383046297535433446893869884, 11.02339945853649596812317093219
