| L(s) = 1 | + (0.904 − 0.159i)2-s + (−0.464 − 0.553i)3-s + (−2.96 + 1.07i)4-s + (0.295 + 0.107i)5-s + (−0.508 − 0.426i)6-s + (−0.328 + 0.569i)7-s + (−5.69 + 3.28i)8-s + (1.47 − 8.34i)9-s + (0.284 + 0.0501i)10-s + (5.74 + 9.94i)11-s + (1.97 + 1.14i)12-s + (8.87 − 10.5i)13-s + (−0.206 + 0.567i)14-s + (−0.0777 − 0.213i)15-s + (5.04 − 4.23i)16-s + (2.87 + 16.3i)17-s + ⋯ |
| L(s) = 1 | + (0.452 − 0.0797i)2-s + (−0.154 − 0.184i)3-s + (−0.741 + 0.269i)4-s + (0.0590 + 0.0215i)5-s + (−0.0848 − 0.0711i)6-s + (−0.0469 + 0.0813i)7-s + (−0.711 + 0.410i)8-s + (0.163 − 0.927i)9-s + (0.0284 + 0.00501i)10-s + (0.522 + 0.904i)11-s + (0.164 + 0.0951i)12-s + (0.682 − 0.813i)13-s + (−0.0147 + 0.0405i)14-s + (−0.00518 − 0.0142i)15-s + (0.315 − 0.264i)16-s + (0.169 + 0.959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.885125 - 0.0418037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.885125 - 0.0418037i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (18.1 + 5.47i)T \) |
| good | 2 | \( 1 + (-0.904 + 0.159i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (0.464 + 0.553i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (-0.295 - 0.107i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (0.328 - 0.569i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.74 - 9.94i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.87 + 10.5i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-2.87 - 16.3i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (32.5 - 11.8i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-23.6 - 4.17i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (-36.6 - 21.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 38.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-19.9 - 23.8i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (23.3 + 8.50i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-0.261 + 1.48i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (23.9 + 65.7i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-60.4 + 10.6i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-7.01 + 2.55i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (4.76 + 0.839i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-6.72 + 18.4i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (50.3 - 42.2i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (32.2 + 38.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (19.0 - 33.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (85.8 - 102. i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (79.6 - 14.0i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−17.90632792878226515946219383797, −17.54466012867784821242614004176, −15.50808949290817193370930696175, −14.33130474175726572597931286898, −12.88981760214387421387792387686, −12.04665200640286573400428642307, −9.945150255058806959779452007930, −8.380151860999356558943800432162, −6.15860181736768261401115256588, −4.01832006244486933641761941779,
4.29342940953308004538746109581, 6.06660408216617329579612142416, 8.494601405815437823900857074649, 10.05210880060430761648114298502, 11.68848369883703645258101762759, 13.48658838483388259388659960450, 14.10879644932805855804378828490, 15.80361697013986644074638216217, 16.95448292638262484701579456003, 18.52762841005297443402200355601
