| L(s) = 1 | + (−6.65 − 9.15i)2-s + (39.9 + 122. i)3-s + (−39.5 + 121. i)4-s + (594. + 431. i)5-s + (859. − 1.18e3i)6-s + (−4.36e3 − 1.41e3i)7-s + (1.37e3 − 447. i)8-s + (−8.21e3 + 5.97e3i)9-s − 8.30e3i·10-s + (9.16e3 + 1.14e4i)11-s − 1.65e4·12-s + (−1.03e4 − 1.41e4i)13-s + (1.60e4 + 4.93e4i)14-s + (−2.93e4 + 9.03e4i)15-s + (−1.32e4 − 9.63e3i)16-s + (−6.13e4 + 8.44e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.493 + 1.51i)3-s + (−0.154 + 0.475i)4-s + (0.950 + 0.690i)5-s + (0.663 − 0.913i)6-s + (−1.81 − 0.590i)7-s + (0.336 − 0.109i)8-s + (−1.25 + 0.910i)9-s − 0.830i·10-s + (0.626 + 0.779i)11-s − 0.798·12-s + (−0.360 − 0.496i)13-s + (0.417 + 1.28i)14-s + (−0.579 + 1.78i)15-s + (−0.202 − 0.146i)16-s + (−0.734 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.464175 + 1.06100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.464175 + 1.06100i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.65 + 9.15i)T \) |
| 11 | \( 1 + (-9.16e3 - 1.14e4i)T \) |
| good | 3 | \( 1 + (-39.9 - 122. i)T + (-5.30e3 + 3.85e3i)T^{2} \) |
| 5 | \( 1 + (-594. - 431. i)T + (1.20e5 + 3.71e5i)T^{2} \) |
| 7 | \( 1 + (4.36e3 + 1.41e3i)T + (4.66e6 + 3.38e6i)T^{2} \) |
| 13 | \( 1 + (1.03e4 + 1.41e4i)T + (-2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (6.13e4 - 8.44e4i)T + (-2.15e9 - 6.63e9i)T^{2} \) |
| 19 | \( 1 + (5.92e4 - 1.92e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + 8.89e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-1.20e6 - 3.91e5i)T + (4.04e11 + 2.94e11i)T^{2} \) |
| 31 | \( 1 + (2.48e5 - 1.80e5i)T + (2.63e11 - 8.11e11i)T^{2} \) |
| 37 | \( 1 + (3.46e5 - 1.06e6i)T + (-2.84e12 - 2.06e12i)T^{2} \) |
| 41 | \( 1 + (-2.50e6 + 8.13e5i)T + (6.45e12 - 4.69e12i)T^{2} \) |
| 43 | \( 1 - 1.14e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-5.20e5 - 1.60e6i)T + (-1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-8.28e6 + 6.02e6i)T + (1.92e13 - 5.92e13i)T^{2} \) |
| 59 | \( 1 + (4.88e6 - 1.50e7i)T + (-1.18e14 - 8.63e13i)T^{2} \) |
| 61 | \( 1 + (-1.93e6 + 2.66e6i)T + (-5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + 2.66e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-9.83e6 - 7.14e6i)T + (1.99e14 + 6.14e14i)T^{2} \) |
| 73 | \( 1 + (-1.77e7 - 5.77e6i)T + (6.52e14 + 4.74e14i)T^{2} \) |
| 79 | \( 1 + (8.96e6 + 1.23e7i)T + (-4.68e14 + 1.44e15i)T^{2} \) |
| 83 | \( 1 + (1.21e6 - 1.66e6i)T + (-6.95e14 - 2.14e15i)T^{2} \) |
| 89 | \( 1 + 4.70e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-7.58e7 + 5.50e7i)T + (2.42e15 - 7.45e15i)T^{2} \) |
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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
−16.62281622753641541136887660337, −15.42893027352663335639064867665, −14.13523751142830992809621530234, −12.75222927966099746938771593043, −10.33676205403045578654155857165, −10.16213142889988330632720532043, −9.063539437530684744049143448050, −6.53156933668132551847315915445, −4.02945037971812227280019155371, −2.74784592224096037673525833086,
0.59331468203182017565431234366, 2.41115486835344079816656531506, 6.04099365233044948052414680431, 6.82666403120980680830247652541, 8.749369279682615155329282813438, 9.502094343095021489910434734418, 12.20667289545969937414149436288, 13.28413689821499871801470815019, 13.96972197985772912481887536969, 15.93941248627271693336722876468
