| L(s) = 1 | + (0.223 + 1.39i)2-s + (0.855 + 0.311i)3-s + (−1.90 + 0.623i)4-s + (−0.00805 − 0.0456i)5-s + (−0.243 + 1.26i)6-s + (1.20 + 0.698i)7-s + (−1.29 − 2.51i)8-s + (−1.66 − 1.39i)9-s + (0.0620 − 0.0214i)10-s + (1.13 − 0.655i)11-s + (−1.82 − 0.0581i)12-s + (−0.681 − 1.87i)13-s + (−0.705 + 1.84i)14-s + (0.00733 − 0.0416i)15-s + (3.22 − 2.37i)16-s + (−0.910 + 0.763i)17-s + ⋯ |
| L(s) = 1 | + (0.157 + 0.987i)2-s + (0.494 + 0.179i)3-s + (−0.950 + 0.311i)4-s + (−0.00360 − 0.0204i)5-s + (−0.0995 + 0.516i)6-s + (0.457 + 0.263i)7-s + (−0.457 − 0.888i)8-s + (−0.554 − 0.465i)9-s + (0.0196 − 0.00678i)10-s + (0.342 − 0.197i)11-s + (−0.525 − 0.0167i)12-s + (−0.189 − 0.519i)13-s + (−0.188 + 0.493i)14-s + (0.00189 − 0.0107i)15-s + (0.805 − 0.592i)16-s + (−0.220 + 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.829406 + 0.643798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.829406 + 0.643798i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.223 - 1.39i)T \) |
| 19 | \( 1 + (-4.35 - 0.189i)T \) |
| good | 3 | \( 1 + (-0.855 - 0.311i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.00805 + 0.0456i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.20 - 0.698i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.13 + 0.655i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.681 + 1.87i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.910 - 0.763i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (8.42 + 1.48i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.45 - 4.11i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.06 - 5.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.41iT - 37T^{2} \) |
| 41 | \( 1 + (2.39 - 6.59i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.406 + 0.0716i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.36 - 6.39i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-5.17 - 0.913i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.98 + 7.53i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.12 + 12.0i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.40 - 7.05i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.06 + 6.02i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (6.09 + 2.21i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 3.76i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.635 - 0.367i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.73 + 12.9i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.45 - 6.50i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−14.56258590932722976080250302843, −14.21234021849626501702792989856, −12.81741791203464083746747524427, −11.67443100555230889394311677314, −9.913973789787350818419701076797, −8.772436537706380495350803943274, −7.963345359536257235244066591642, −6.43562746169851248938720464217, −5.12886296190883871858429141291, −3.45461323557222292809641975874,
2.14983598692062693497490721023, 3.91716143632953141335071926658, 5.46408654182023905010931323882, 7.57519122577656645513177668397, 8.798897977646620701996825988070, 9.870592229202261700187580456036, 11.19458714992723394637364500034, 11.93799615025788423269906386578, 13.34121847574722176381611967421, 14.05725374921432461673619605495
