| L(s) = 1 | + (−2.84 − 0.501i)2-s + (3.28 − 3.91i)3-s + (4.07 + 1.48i)4-s + (−3.00 + 1.09i)5-s + (−11.3 + 9.48i)6-s + (3.87 + 6.70i)7-s + (−0.853 − 0.492i)8-s + (−2.96 − 16.8i)9-s + (9.08 − 1.60i)10-s + (−3.45 + 5.98i)11-s + (19.2 − 11.0i)12-s + (3.92 + 4.67i)13-s + (−7.64 − 21.0i)14-s + (−5.58 + 15.3i)15-s + (−11.1 − 9.33i)16-s + (0.218 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−1.42 − 0.250i)2-s + (1.09 − 1.30i)3-s + (1.01 + 0.371i)4-s + (−0.600 + 0.218i)5-s + (−1.88 + 1.58i)6-s + (0.553 + 0.958i)7-s + (−0.106 − 0.0615i)8-s + (−0.329 − 1.87i)9-s + (0.908 − 0.160i)10-s + (−0.313 + 0.543i)11-s + (1.60 − 0.924i)12-s + (0.301 + 0.359i)13-s + (−0.546 − 1.50i)14-s + (−0.372 + 1.02i)15-s + (−0.695 − 0.583i)16-s + (0.0128 − 0.0728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.512420 - 0.286804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.512420 - 0.286804i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (12.9 + 13.9i)T \) |
| good | 2 | \( 1 + (2.84 + 0.501i)T + (3.75 + 1.36i)T^{2} \) |
| 3 | \( 1 + (-3.28 + 3.91i)T + (-1.56 - 8.86i)T^{2} \) |
| 5 | \( 1 + (3.00 - 1.09i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-3.87 - 6.70i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.45 - 5.98i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-3.92 - 4.67i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-0.218 + 1.23i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (1.77 + 0.647i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-11.5 + 2.04i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (24.9 - 14.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 43.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-19.6 + 23.3i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (5.57 - 2.02i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-13.9 - 79.0i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (-26.8 + 73.6i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (74.2 + 13.0i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-26.7 - 9.72i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-126. + 22.3i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (1.86 + 5.11i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-3.22 - 2.70i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (57.2 - 68.2i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-10.4 - 18.1i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (94.4 + 112. i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-68.4 - 12.0i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−18.37906482709185152146595092842, −17.62511053450663478354219119303, −15.58396295057909844918137805948, −14.27230707615780527055266333245, −12.62298293666636854439218441753, −11.27817673474638172503156450990, −9.160289843186077973989152486560, −8.236158146884272357560343718549, −7.18932761619157437526033486191, −2.16256056132298264774280904277,
4.09724206788381690037057893811, 7.85106244590997063393480296330, 8.564888976276567385575191209726, 10.03108164558663278526080015789, 10.89812034901162709327059162101, 13.73295326110993913951577055025, 15.14033872450998114141815961400, 16.16259310347314101754428257156, 17.01517690472622516748065355958, 18.66345926751299061565767690120
