| L(s) = 1 | + (−5.01 + 5.97i)2-s + (4.42 + 12.1i)3-s + (−7.80 − 44.2i)4-s + (−1.37 + 7.82i)5-s + (−94.9 − 34.5i)6-s + (−2.12 − 3.68i)7-s + (195. + 112. i)8-s + (−66.2 + 55.5i)9-s + (−39.8 − 47.4i)10-s + (−34.8 + 60.3i)11-s + (503. − 290. i)12-s + (−46.7 + 128. i)13-s + (32.7 + 5.77i)14-s + (−101. + 17.8i)15-s + (−980. + 356. i)16-s + (152. + 127. i)17-s + ⋯ |
| L(s) = 1 | + (−1.25 + 1.49i)2-s + (0.491 + 1.35i)3-s + (−0.487 − 2.76i)4-s + (−0.0551 + 0.312i)5-s + (−2.63 − 0.959i)6-s + (−0.0434 − 0.0752i)7-s + (3.05 + 1.76i)8-s + (−0.817 + 0.685i)9-s + (−0.398 − 0.474i)10-s + (−0.288 + 0.499i)11-s + (3.49 − 2.01i)12-s + (−0.276 + 0.760i)13-s + (0.166 + 0.0294i)14-s + (−0.449 + 0.0793i)15-s + (−3.82 + 1.39i)16-s + (0.526 + 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0514876 - 0.706847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0514876 - 0.706847i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-240. + 269. i)T \) |
| good | 2 | \( 1 + (5.01 - 5.97i)T + (-2.77 - 15.7i)T^{2} \) |
| 3 | \( 1 + (-4.42 - 12.1i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (1.37 - 7.82i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (2.12 + 3.68i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (34.8 - 60.3i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (46.7 - 128. i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-152. - 127. i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (57.8 + 328. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-624. - 744. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-43.5 + 25.1i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 51.5iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (338. + 931. i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (-285. + 1.62e3i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (2.68e3 - 2.25e3i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-2.85e3 + 503. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (-776. + 925. i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (888. + 5.03e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-2.04e3 - 2.43e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (6.51e3 + 1.14e3i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-3.99e3 + 1.45e3i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (1.52e3 + 4.18e3i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (444. + 770. i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.97e3 - 5.43e3i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (3.12e3 - 3.72e3i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
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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
−18.11725186550358089383373405703, −16.79790129339463805042778344232, −15.94522779670908664701722841721, −14.99165851809491479428388745230, −14.20513701184700423498445847715, −10.65408003184295442329608338692, −9.708751534555555471243008382046, −8.642686768889211195150538366567, −6.99721181853470969881149667057, −4.93238493076033256586372185064,
1.01449727075772186044327566718, 2.86502394083837241889570338924, 7.57567625416940977586415177757, 8.461527033567454258022942979642, 10.03494808274857775169419023784, 11.73774052514968287874427945690, 12.65384467364652984646322985945, 13.65991925907542356192893959132, 16.45527033663601497335214946544, 17.84261558020840476572626446682
