| 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
−11.02339945853649596812317093219, −10.67383046297535433446893869884, −9.290689355167905841005241994652, −7.71404140957434722408918877107, −6.99221661031242577290521465415, −6.34870512346136221462771424723, −5.26778866858226273876336658494, −4.06255184470114850114373960441, −3.60193691633532909002231519036, −0.70549669352861379554312781107,
1.65426088975420531367870868062, 3.80915940279879575389468187730, 4.34509180484814731934738985442, 5.24153917125559302168875385571, 6.14168089665337777800161449300, 7.56575294746097839153664912300, 8.649869321524629623754461001344, 9.278940796239506481849682872983, 10.95639930640416233643610172617, 11.60090750269382949133414638781
