| L(s) = 1 | + (406. + 234. i)2-s + (1.80e4 − 7.89e3i)3-s + (−2.09e4 − 3.63e4i)4-s + (−2.94e6 + 1.69e6i)5-s + (9.17e6 + 1.02e6i)6-s + (2.46e7 − 4.26e7i)7-s − 1.42e8i·8-s + (2.62e8 − 2.84e8i)9-s − 1.59e9·10-s + (1.16e9 + 6.74e8i)11-s + (−6.65e8 − 4.89e8i)12-s + (−8.99e9 − 1.55e10i)13-s + (2.00e10 − 1.15e10i)14-s + (−3.96e10 + 5.38e10i)15-s + (2.79e10 − 4.84e10i)16-s − 8.08e10i·17-s + ⋯ |
| L(s) = 1 | + (0.793 + 0.458i)2-s + (0.916 − 0.401i)3-s + (−0.0800 − 0.138i)4-s + (−1.50 + 0.869i)5-s + (0.910 + 0.101i)6-s + (0.609 − 1.05i)7-s − 1.06i·8-s + (0.678 − 0.734i)9-s − 1.59·10-s + (0.495 + 0.286i)11-s + (−0.128 − 0.0948i)12-s + (−0.847 − 1.46i)13-s + (0.968 − 0.559i)14-s + (−1.03 + 1.40i)15-s + (0.407 − 0.705i)16-s − 0.681i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.87686 - 1.65206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87686 - 1.65206i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.80e4 + 7.89e3i)T \) |
| good | 2 | \( 1 + (-406. - 234. i)T + (1.31e5 + 2.27e5i)T^{2} \) |
| 5 | \( 1 + (2.94e6 - 1.69e6i)T + (1.90e12 - 3.30e12i)T^{2} \) |
| 7 | \( 1 + (-2.46e7 + 4.26e7i)T + (-8.14e14 - 1.41e15i)T^{2} \) |
| 11 | \( 1 + (-1.16e9 - 6.74e8i)T + (2.77e18 + 4.81e18i)T^{2} \) |
| 13 | \( 1 + (8.99e9 + 1.55e10i)T + (-5.62e19 + 9.73e19i)T^{2} \) |
| 17 | \( 1 + 8.08e10iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 4.20e10T + 1.04e23T^{2} \) |
| 23 | \( 1 + (1.91e12 - 1.10e12i)T + (1.62e24 - 2.80e24i)T^{2} \) |
| 29 | \( 1 + (-1.29e13 - 7.46e12i)T + (1.05e26 + 1.82e26i)T^{2} \) |
| 31 | \( 1 + (-6.04e12 - 1.04e13i)T + (-3.49e26 + 6.05e26i)T^{2} \) |
| 37 | \( 1 - 3.75e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + (2.99e14 - 1.72e14i)T + (5.35e28 - 9.28e28i)T^{2} \) |
| 43 | \( 1 + (-2.00e14 + 3.47e14i)T + (-1.26e29 - 2.18e29i)T^{2} \) |
| 47 | \( 1 + (-9.94e14 - 5.74e14i)T + (6.26e29 + 1.08e30i)T^{2} \) |
| 53 | \( 1 + 9.69e14iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (-6.01e15 + 3.47e15i)T + (3.75e31 - 6.49e31i)T^{2} \) |
| 61 | \( 1 + (8.21e15 - 1.42e16i)T + (-6.83e31 - 1.18e32i)T^{2} \) |
| 67 | \( 1 + (6.22e15 + 1.07e16i)T + (-3.70e32 + 6.41e32i)T^{2} \) |
| 71 | \( 1 + 1.58e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 5.96e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (-2.71e16 + 4.69e16i)T + (-7.18e33 - 1.24e34i)T^{2} \) |
| 83 | \( 1 + (-5.61e16 - 3.24e16i)T + (1.74e34 + 3.02e34i)T^{2} \) |
| 89 | \( 1 + 1.44e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-2.08e17 + 3.60e17i)T + (-2.88e35 - 5.00e35i)T^{2} \) |
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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
−15.62560912833252443714484959054, −14.77016713060591016005049716263, −13.89795760776642442235222693492, −12.21227173511863511984109996283, −10.27229030984801819657500481505, −7.81617559184214002427697478514, −7.02510547668882325688023890539, −4.40777988123296789291262110523, −3.28076527104681426837914621319, −0.66561231308378429223707589510,
2.17435681151162332264208177900, 3.93465976381238459206033100040, 4.68801249607893250405715511726, 8.071687243457699355444994343735, 8.866189092667124366548839614570, 11.65051359590552721626124454108, 12.37053249925191087231063252325, 14.16528267544654476240351167803, 15.26775383765883073190991825896, 16.66649859950119423495878778234
