| L(s) = 1 | + (689. − 397. i)2-s + (−1.94e4 − 2.84e3i)3-s + (1.85e5 − 3.21e5i)4-s + (−7.68e5 − 4.43e5i)5-s + (−1.45e7 + 5.78e6i)6-s + (−1.10e7 − 1.90e7i)7-s − 8.69e7i·8-s + (3.71e8 + 1.10e8i)9-s − 7.06e8·10-s + (−2.62e9 + 1.51e9i)11-s + (−4.53e9 + 5.73e9i)12-s + (−8.49e9 + 1.47e10i)13-s + (−1.51e10 − 8.77e9i)14-s + (1.37e10 + 1.08e10i)15-s + (1.40e10 + 2.43e10i)16-s − 8.22e10i·17-s + ⋯ |
| L(s) = 1 | + (1.34 − 0.777i)2-s + (−0.989 − 0.144i)3-s + (0.708 − 1.22i)4-s + (−0.393 − 0.227i)5-s + (−1.44 + 0.574i)6-s + (−0.273 − 0.473i)7-s − 0.648i·8-s + (0.958 + 0.286i)9-s − 0.706·10-s + (−1.11 + 0.642i)11-s + (−0.878 + 1.11i)12-s + (−0.801 + 1.38i)13-s + (−0.735 − 0.424i)14-s + (0.356 + 0.281i)15-s + (0.204 + 0.354i)16-s − 0.693i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.00894391 + 0.0136096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00894391 + 0.0136096i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.94e4 + 2.84e3i)T \) |
| good | 2 | \( 1 + (-689. + 397. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (7.68e5 + 4.43e5i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (1.10e7 + 1.90e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (2.62e9 - 1.51e9i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (8.49e9 - 1.47e10i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 + 8.22e10iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 8.50e10T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-2.25e11 - 1.30e11i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (1.34e13 - 7.77e12i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (-1.38e13 + 2.40e13i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 + 2.24e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + (2.26e14 + 1.30e14i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (1.11e14 + 1.93e14i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (-9.54e14 + 5.51e14i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 - 8.15e14iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (1.42e16 + 8.22e15i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (-6.40e15 - 1.10e16i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (-9.87e15 + 1.71e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 - 8.38e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 6.27e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (-8.23e16 - 1.42e17i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (1.57e17 - 9.07e16i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 + 2.83e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-4.48e17 - 7.76e17i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−15.54810051143932891478436869900, −13.75947072901762853229094093497, −12.52571390324209390245695175418, −11.61888952818295460006063212429, −10.22693027178722777702888596790, −7.07300403466957890660247557381, −5.20451870855453761190835529768, −4.17373799565907082043517932081, −2.06717913451788828455766154407, −0.00410941732063235789686410745,
3.27238495159059449260892122641, 5.04693235181943007987145447130, 5.97729137340935615647933817503, 7.58499609652404587023668889761, 10.50206484795357529163989147337, 12.21539118834108097691085968921, 13.17767004075245626963588049330, 15.16646407201879440139972338463, 15.76336501787316909274920497337, 17.22752909127222933288769960446
