| L(s) = 1 | + (−462. + 266. i)2-s + (1.89e4 + 5.17e3i)3-s + (1.13e4 − 1.96e4i)4-s + (1.29e6 + 7.45e5i)5-s + (−1.01e7 + 2.67e6i)6-s + (2.69e7 + 4.66e7i)7-s − 1.27e8i·8-s + (3.33e8 + 1.96e8i)9-s − 7.95e8·10-s + (2.73e9 − 1.58e9i)11-s + (3.17e8 − 3.14e8i)12-s + (8.99e9 − 1.55e10i)13-s + (−2.49e10 − 1.43e10i)14-s + (2.06e10 + 2.08e10i)15-s + (3.70e10 + 6.42e10i)16-s + 2.10e11i·17-s + ⋯ |
| L(s) = 1 | + (−0.902 + 0.521i)2-s + (0.964 + 0.262i)3-s + (0.0433 − 0.0750i)4-s + (0.660 + 0.381i)5-s + (−1.00 + 0.265i)6-s + (0.667 + 1.15i)7-s − 0.952i·8-s + (0.861 + 0.507i)9-s − 0.795·10-s + (1.16 − 0.670i)11-s + (0.0615 − 0.0610i)12-s + (0.848 − 1.46i)13-s + (−1.20 − 0.695i)14-s + (0.537 + 0.541i)15-s + (0.539 + 0.934i)16-s + 1.77i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.32391 + 1.56427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32391 + 1.56427i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.89e4 - 5.17e3i)T \) |
| good | 2 | \( 1 + (462. - 266. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (-1.29e6 - 7.45e5i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (-2.69e7 - 4.66e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (-2.73e9 + 1.58e9i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (-8.99e9 + 1.55e10i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 - 2.10e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 1.84e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-4.12e11 - 2.38e11i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (2.02e12 - 1.16e12i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (1.04e13 - 1.81e13i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 + 1.06e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + (-2.03e14 - 1.17e14i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (1.43e14 + 2.48e14i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (5.89e14 - 3.40e14i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 + 1.11e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (-2.76e15 - 1.59e15i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (5.11e15 + 8.86e15i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (-1.85e16 + 3.21e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 - 2.96e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 6.02e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (8.11e16 + 1.40e17i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (5.31e16 - 3.06e16i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 + 4.07e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-8.80e16 - 1.52e17i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−17.32857692089861824613337340271, −15.61502211920900521252050215134, −14.57148734129407896187134471806, −12.89050202939920691102399369760, −10.45316220645334582151666059768, −8.865602289328139421267817344812, −8.238755861953367286270418532772, −6.12725891192067888870808394037, −3.49733273924602456017447275595, −1.60398791641024839490340043381,
1.10300570954092335896213767251, 1.89335711805718378622016633265, 4.32187626181057700135876602924, 7.12378455652238798467426413399, 8.890133974622790779636256092522, 9.687033068482169633734610828205, 11.44207162478315377619433370668, 13.68349165988079079508281236493, 14.38786728323960168354858735490, 16.81434975355772663390447983770
