| L(s) = 1 | + (−307. + 177. i)2-s + (3.07e3 + 1.94e4i)3-s + (−6.81e4 + 1.18e5i)4-s + (−9.96e5 − 5.75e5i)5-s + (−4.39e6 − 5.42e6i)6-s + (−8.05e6 − 1.39e7i)7-s − 1.41e8i·8-s + (−3.68e8 + 1.19e8i)9-s + 4.08e8·10-s + (−8.24e8 + 4.76e8i)11-s + (−2.50e9 − 9.61e8i)12-s + (−3.57e9 + 6.18e9i)13-s + (4.95e9 + 2.85e9i)14-s + (8.12e9 − 2.11e10i)15-s + (7.21e9 + 1.24e10i)16-s − 7.71e10i·17-s + ⋯ |
| L(s) = 1 | + (−0.600 + 0.346i)2-s + (0.156 + 0.987i)3-s + (−0.259 + 0.450i)4-s + (−0.510 − 0.294i)5-s + (−0.436 − 0.538i)6-s + (−0.199 − 0.345i)7-s − 1.05i·8-s + (−0.951 + 0.308i)9-s + 0.408·10-s + (−0.349 + 0.201i)11-s + (−0.485 − 0.186i)12-s + (−0.336 + 0.583i)13-s + (0.239 + 0.138i)14-s + (0.211 − 0.550i)15-s + (0.104 + 0.181i)16-s − 0.650i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.458361 - 0.131480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458361 - 0.131480i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.07e3 - 1.94e4i)T \) |
| good | 2 | \( 1 + (307. - 177. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (9.96e5 + 5.75e5i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (8.05e6 + 1.39e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (8.24e8 - 4.76e8i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (3.57e9 - 6.18e9i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 + 7.71e10iT - 1.40e22T^{2} \) |
| 19 | \( 1 - 4.16e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-1.37e12 - 7.92e11i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (-1.27e13 + 7.35e12i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (9.68e12 - 1.67e13i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 + 2.09e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + (-2.91e13 - 1.68e13i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (-1.39e14 - 2.41e14i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (-1.11e15 + 6.43e14i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 + 4.94e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (7.13e15 + 4.12e15i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (4.87e15 + 8.44e15i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (1.90e16 - 3.30e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 + 6.62e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 8.63e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (4.93e16 + 8.54e16i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (-2.27e17 + 1.31e17i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 + 1.08e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (5.46e17 + 9.45e17i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−16.43061812734121941438503017779, −15.68705605707347400301232851862, −13.80979562455858820249967760801, −11.91204152459720978707249773255, −10.00423586423275904878347119910, −8.825748579708225776082545569865, −7.37640664170202138395815458191, −4.75873426779030435599334086938, −3.34605928098256924194631153592, −0.25439567415549460879326152165,
1.12630709937736271407592205243, 2.82177948109881636071202348796, 5.61703394756026533577449879279, 7.54807657610840121744070653061, 8.948906128439042373934834767582, 10.74822053247406118055542880467, 12.22364757803115935494091159824, 13.84006137238638454067876841752, 15.23920680382064691802835732405, 17.39681885548693948765954124061
