L(s) = 1 | + (0.579 + 0.210i)2-s + (1.63 + 9.27i)3-s + (−5.83 − 4.89i)4-s + (9.88 − 8.29i)5-s + (−1.00 + 5.71i)6-s + (6.05 − 10.4i)7-s + (−4.81 − 8.33i)8-s + (−57.9 + 21.0i)9-s + (7.46 − 2.71i)10-s + (1.51 + 2.61i)11-s + (35.8 − 62.1i)12-s + (−2.02 + 11.4i)13-s + (5.71 − 4.79i)14-s + (93.0 + 78.0i)15-s + (9.55 + 54.1i)16-s + (−104. − 38.0i)17-s + ⋯ |
L(s) = 1 | + (0.204 + 0.0745i)2-s + (0.314 + 1.78i)3-s + (−0.729 − 0.612i)4-s + (0.883 − 0.741i)5-s + (−0.0685 + 0.388i)6-s + (0.327 − 0.566i)7-s + (−0.212 − 0.368i)8-s + (−2.14 + 0.781i)9-s + (0.236 − 0.0859i)10-s + (0.0414 + 0.0717i)11-s + (0.863 − 1.49i)12-s + (−0.0432 + 0.245i)13-s + (0.109 − 0.0916i)14-s + (1.60 + 1.34i)15-s + (0.149 + 0.846i)16-s + (−1.49 − 0.542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.11107 + 0.451613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11107 + 0.451613i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + (-82.6 - 5.09i)T \) |
good | 2 | \( 1 + (-0.579 - 0.210i)T + (6.12 + 5.14i)T^{2} \) |
| 3 | \( 1 + (-1.63 - 9.27i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (-9.88 + 8.29i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-6.05 + 10.4i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-1.51 - 2.61i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (2.02 - 11.4i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (104. + 38.0i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (0.581 + 0.487i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (96.6 - 35.1i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (59.9 - 103. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (17.2 + 97.7i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-88.1 + 73.9i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-298. + 108. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (244. + 205. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-225. - 82.0i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (166. + 139. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (510. - 185. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-166. + 139. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (99.2 + 562. i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-185. - 1.05e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (315. - 546. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (23.4 - 132. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-859. - 312. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−17.87738343950880896643194610084, −16.74660068488962049713388767220, −15.55741824976371235771066126492, −14.31437017640345369088031410350, −13.46834804406925733367034118405, −10.90114475097715645673970018449, −9.627613425538338900979465487093, −8.990005529828433178660473681652, −5.37910527992928316945820457893, −4.32819504465645121356367857857,
2.48099309715001762746499871625, 6.05004923023977833302968219967, 7.64933940408179660916685891502, 9.031602240867070571421724529449, 11.55726474177970449670943983123, 12.91848157435819790628321412440, 13.62886766726311204760573949067, 14.66982722899333181059117633242, 17.36255154994526994279959887362, 18.06008442891240130188924874527