| L(s) = 1 | + (1.25 − 1.49i)2-s + (1.82 + 5.01i)3-s + (10.4 + 59.2i)4-s + (−2.44 + 13.8i)5-s + (9.77 + 3.55i)6-s + (212. + 368. i)7-s + (209. + 121. i)8-s + (536. − 450. i)9-s + (17.6 + 21.0i)10-s + (−1.22e3 + 2.12e3i)11-s + (−278. + 160. i)12-s + (1.21e3 − 3.33e3i)13-s + (816. + 143. i)14-s + (−74.1 + 13.0i)15-s + (−3.17e3 + 1.15e3i)16-s + (−2.36e3 − 1.98e3i)17-s + ⋯ |
| L(s) = 1 | + (0.156 − 0.186i)2-s + (0.0676 + 0.185i)3-s + (0.163 + 0.926i)4-s + (−0.0195 + 0.111i)5-s + (0.0452 + 0.0164i)6-s + (0.619 + 1.07i)7-s + (0.409 + 0.236i)8-s + (0.736 − 0.617i)9-s + (0.0176 + 0.0210i)10-s + (−0.921 + 1.59i)11-s + (−0.161 + 0.0930i)12-s + (0.552 − 1.51i)13-s + (0.297 + 0.0524i)14-s + (−0.0219 + 0.00387i)15-s + (−0.775 + 0.282i)16-s + (−0.480 − 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.55474 + 0.808312i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55474 + 0.808312i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (586. + 6.83e3i)T \) |
| good | 2 | \( 1 + (-1.25 + 1.49i)T + (-11.1 - 63.0i)T^{2} \) |
| 3 | \( 1 + (-1.82 - 5.01i)T + (-558. + 468. i)T^{2} \) |
| 5 | \( 1 + (2.44 - 13.8i)T + (-1.46e4 - 5.34e3i)T^{2} \) |
| 7 | \( 1 + (-212. - 368. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.22e3 - 2.12e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.21e3 + 3.33e3i)T + (-3.69e6 - 3.10e6i)T^{2} \) |
| 17 | \( 1 + (2.36e3 + 1.98e3i)T + (4.19e6 + 2.37e7i)T^{2} \) |
| 23 | \( 1 + (1.20e3 + 6.85e3i)T + (-1.39e8 + 5.06e7i)T^{2} \) |
| 29 | \( 1 + (-1.98e4 - 2.36e4i)T + (-1.03e8 + 5.85e8i)T^{2} \) |
| 31 | \( 1 + (-2.12e4 + 1.22e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 6.36e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (2.89e4 + 7.95e4i)T + (-3.63e9 + 3.05e9i)T^{2} \) |
| 43 | \( 1 + (8.45e3 - 4.79e4i)T + (-5.94e9 - 2.16e9i)T^{2} \) |
| 47 | \( 1 + (-5.40e4 + 4.53e4i)T + (1.87e9 - 1.06e10i)T^{2} \) |
| 53 | \( 1 + (7.08e4 - 1.24e4i)T + (2.08e10 - 7.58e9i)T^{2} \) |
| 59 | \( 1 + (1.68e5 - 2.00e5i)T + (-7.32e9 - 4.15e10i)T^{2} \) |
| 61 | \( 1 + (-2.00e4 - 1.13e5i)T + (-4.84e10 + 1.76e10i)T^{2} \) |
| 67 | \( 1 + (-8.59e4 - 1.02e5i)T + (-1.57e10 + 8.90e10i)T^{2} \) |
| 71 | \( 1 + (4.79e5 + 8.45e4i)T + (1.20e11 + 4.38e10i)T^{2} \) |
| 73 | \( 1 + (-1.29e5 + 4.73e4i)T + (1.15e11 - 9.72e10i)T^{2} \) |
| 79 | \( 1 + (-88.4 - 243. i)T + (-1.86e11 + 1.56e11i)T^{2} \) |
| 83 | \( 1 + (-8.60e3 - 1.49e4i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-1.13e5 + 3.12e5i)T + (-3.80e11 - 3.19e11i)T^{2} \) |
| 97 | \( 1 + (6.53e5 - 7.79e5i)T + (-1.44e11 - 8.20e11i)T^{2} \) |
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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
−17.84153766796612890978147451806, −15.71863878931030308800415426014, −15.15003953109759001026054319461, −12.96657024242973450434924363832, −12.25332867318609019054438215065, −10.60720263345786500763158145958, −8.755069422904496429021752472186, −7.26761059013759364248210669542, −4.79883990640595919628856926229, −2.63402200822665881731716851141,
1.33237784040001635280654932243, 4.56745598510436730096942002557, 6.45756291502774430565704772410, 8.155285008448507522125430741217, 10.28570230305308729042443805874, 11.20730704734221339120757299278, 13.56525383776539294693121004514, 14.01324072625150125928628674108, 15.79779071870284766975767268717, 16.68744993316765325913441386820
