| L(s) = 1 | + (4.43 − 0.781i)2-s + (6.29 + 7.50i)3-s + (4.01 − 1.46i)4-s + (−26.3 − 9.57i)5-s + (33.7 + 28.3i)6-s + (37.8 − 65.6i)7-s + (−45.7 + 26.4i)8-s + (−2.60 + 14.7i)9-s + (−124. − 21.8i)10-s + (46.3 + 80.2i)11-s + (36.2 + 20.9i)12-s + (−105. + 125. i)13-s + (116. − 320. i)14-s + (−93.8 − 257. i)15-s + (−234. + 196. i)16-s + (27.7 + 157. i)17-s + ⋯ |
| L(s) = 1 | + (1.10 − 0.195i)2-s + (0.699 + 0.833i)3-s + (0.250 − 0.0912i)4-s + (−1.05 − 0.383i)5-s + (0.938 + 0.787i)6-s + (0.772 − 1.33i)7-s + (−0.714 + 0.412i)8-s + (−0.0321 + 0.182i)9-s + (−1.24 − 0.218i)10-s + (0.382 + 0.663i)11-s + (0.251 + 0.145i)12-s + (−0.623 + 0.742i)13-s + (0.595 − 1.63i)14-s + (−0.417 − 1.14i)15-s + (−0.915 + 0.768i)16-s + (0.0959 + 0.544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.04264 + 0.167134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04264 + 0.167134i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-360. - 12.4i)T \) |
| good | 2 | \( 1 + (-4.43 + 0.781i)T + (15.0 - 5.47i)T^{2} \) |
| 3 | \( 1 + (-6.29 - 7.50i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (26.3 + 9.57i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (-37.8 + 65.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-46.3 - 80.2i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (105. - 125. i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-27.7 - 157. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-57.4 + 20.8i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-257. - 45.4i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (1.55e3 + 897. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.74e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-407. - 485. i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-2.48e3 - 905. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (700. - 3.97e3i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (929. + 2.55e3i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (3.14e3 - 554. i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (-2.40e3 + 874. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-3.17 - 0.559i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-775. + 2.13e3i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (2.43e3 - 2.04e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (3.53e3 + 4.21e3i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (2.87e3 - 4.98e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.04e3 - 2.43e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (1.30e3 - 229. i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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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
−17.56866664415007904774099539665, −16.09453256266575251280793370990, −14.71561549281990312096254047607, −14.21993350740322994887094951576, −12.53715867571944231496882135650, −11.25423527797206743064616766719, −9.355424078919422862223401012385, −7.64544790732761377800440610691, −4.51147536835924796429444314192, −3.87926369333675880329104645914,
3.05450961109981685700836430811, 5.31708513340490835774267396877, 7.40166659286372683517116628757, 8.756810565014295041951620606463, 11.60831610475703701822673911948, 12.50048784419142690142156952892, 13.94912635817627308314289576462, 14.81691699261016001884328937909, 15.75703291675228952377214633950, 18.20571569926119242715050644012
