| L(s) = 1 | + (−12.6 − 21.8i)2-s + (88.4 + 153. i)3-s + (−63.5 + 110. i)4-s + (498. + 864. i)5-s + (2.23e3 − 3.87e3i)6-s − 9.65e3·7-s − 9.72e3·8-s + (−5.82e3 + 1.00e4i)9-s + (1.26e4 − 2.18e4i)10-s − 665.·11-s − 2.25e4·12-s + (−6.95e4 + 1.20e5i)13-s + (1.22e5 + 2.11e5i)14-s + (−8.83e4 + 1.52e5i)15-s + (1.55e5 + 2.69e5i)16-s + (3.23e4 + 5.60e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.558 − 0.967i)2-s + (0.630 + 1.09i)3-s + (−0.124 + 0.215i)4-s + (0.356 + 0.618i)5-s + (0.704 − 1.22i)6-s − 1.52·7-s − 0.839·8-s + (−0.295 + 0.512i)9-s + (0.398 − 0.690i)10-s − 0.0137·11-s − 0.313·12-s + (−0.675 + 1.16i)13-s + (0.849 + 1.47i)14-s + (−0.450 + 0.780i)15-s + (0.593 + 1.02i)16-s + (0.0940 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.385630 + 0.569729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.385630 + 0.569729i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (5.66e5 - 3.97e4i)T \) |
| good | 2 | \( 1 + (12.6 + 21.8i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-88.4 - 153. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-498. - 864. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 9.65e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 665.T + 2.35e9T^{2} \) |
| 13 | \( 1 + (6.95e4 - 1.20e5i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-3.23e4 - 5.60e4i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (6.07e5 - 1.05e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.32e6 - 2.29e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 4.14e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.80e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.07e7 + 1.86e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.09e7 - 1.89e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-1.53e7 + 2.66e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (4.64e7 - 8.04e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.73e7 + 4.73e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-3.40e7 + 5.89e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.18e8 + 2.04e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (6.87e7 + 1.19e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-6.55e7 - 1.13e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.67e8 - 2.89e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.89e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (4.22e8 - 7.31e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-2.69e8 - 4.66e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−16.68378884333506108614402502309, −15.42166608617754487417741816566, −14.31708791194975778922845553034, −12.50790414330172971131341270245, −10.78360785567488250516031525719, −9.763225791968451878611177872542, −9.201080460693204848315552488877, −6.44478577158237762481576248760, −3.70622588768468856399438604352, −2.43877173108410839257104513937,
0.33476375406855182764051616254, 2.73732416017756416380763089334, 6.05160568581920440235166690165, 7.26254637796277993984085167974, 8.459807847158726245803258229534, 9.730801821912903576161578230759, 12.61940312971464704884462285493, 13.04379740527839053106480449334, 14.82381310974746951634359053201, 16.19006887164530847661839261271
