| L(s) = 1 | + (−4.35 − 9.04i)2-s + (15.2 − 31.5i)3-s + (−22.9 + 28.7i)4-s + (−93.5 − 21.3i)5-s − 351.·6-s + 345. i·7-s + (−266. − 60.8i)8-s + (−311. − 391. i)9-s + (214. + 938. i)10-s + (−1.06e3 − 1.33e3i)11-s + (559. + 1.16e3i)12-s + (211. − 927. i)13-s + (3.12e3 − 1.50e3i)14-s + (−2.09e3 + 2.62e3i)15-s + (1.13e3 + 4.96e3i)16-s + (860. + 3.76e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.544 − 1.13i)2-s + (0.563 − 1.16i)3-s + (−0.358 + 0.448i)4-s + (−0.748 − 0.170i)5-s − 1.62·6-s + 1.00i·7-s + (−0.520 − 0.118i)8-s + (−0.427 − 0.536i)9-s + (0.214 + 0.938i)10-s + (−0.801 − 1.00i)11-s + (0.323 + 0.671i)12-s + (0.0963 − 0.422i)13-s + (1.13 − 0.548i)14-s + (−0.621 + 0.779i)15-s + (0.276 + 1.21i)16-s + (0.175 + 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.401980 + 0.469156i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.401980 + 0.469156i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.01e4 + 6.17e4i)T \) |
| good | 2 | \( 1 + (4.35 + 9.04i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-15.2 + 31.5i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (93.5 + 21.3i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 - 345. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (1.06e3 + 1.33e3i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-211. + 927. i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (-860. - 3.76e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (-3.08e3 - 2.45e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (8.84e3 + 1.10e4i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (6.30e3 + 1.30e4i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (-2.61e3 + 1.26e3i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 + 3.13e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (1.03e5 - 4.97e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (4.73e4 - 5.94e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (5.34e4 + 2.34e5i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (-8.37e3 - 3.66e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.10e5 - 2.29e5i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (-4.27e4 + 5.35e4i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (2.56e5 + 2.04e5i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (4.20e5 + 9.60e4i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 - 3.47e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-9.23e5 - 4.44e5i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (-5.25e5 + 1.09e6i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (-2.10e5 - 2.64e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−13.37501489852675922790391413162, −12.39588806820403160550458119798, −11.67327121027651659233470575981, −10.29987746686159015102270748115, −8.583762865101907256320603916049, −8.024512081311408341374209723830, −6.01074802232588682369704748302, −3.19877721354309762003026315229, −1.95161412163665383126443180434, −0.31232288813341823596193292062,
3.43811757808302435908922218863, 4.87121485622073907285001972804, 7.08118880385385180937208506045, 7.87802971799108326884169651822, 9.315237902592341950833279847798, 10.24011248376378844291158065939, 11.78521987230795597779811986797, 13.75622335625798133433647693120, 14.94207223469419174003626278178, 15.64301458084123323624181205753
