| L(s) = 1 | + (−7.01 − 1.23i)2-s + (−2.17 + 2.58i)3-s + (32.6 + 11.8i)4-s + (26.1 − 9.52i)5-s + (18.4 − 15.4i)6-s + (22.6 + 39.2i)7-s + (−115. − 66.5i)8-s + (12.0 + 68.5i)9-s + (−195. + 34.4i)10-s + (93.7 − 162. i)11-s + (−101. + 58.5i)12-s + (94.3 + 112. i)13-s + (−110. − 303. i)14-s + (−32.1 + 88.3i)15-s + (300. + 252. i)16-s + (−51.2 + 290. i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 − 0.309i)2-s + (−0.241 + 0.287i)3-s + (2.03 + 0.741i)4-s + (1.04 − 0.381i)5-s + (0.511 − 0.429i)6-s + (0.462 + 0.801i)7-s + (−1.80 − 1.03i)8-s + (0.149 + 0.846i)9-s + (−1.95 + 0.344i)10-s + (0.774 − 1.34i)11-s + (−0.704 + 0.406i)12-s + (0.558 + 0.665i)13-s + (−0.563 − 1.54i)14-s + (−0.142 + 0.392i)15-s + (1.17 + 0.985i)16-s + (−0.177 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.667553 + 0.0712361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667553 + 0.0712361i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-331. + 144. i)T \) |
| good | 2 | \( 1 + (7.01 + 1.23i)T + (15.0 + 5.47i)T^{2} \) |
| 3 | \( 1 + (2.17 - 2.58i)T + (-14.0 - 79.7i)T^{2} \) |
| 5 | \( 1 + (-26.1 + 9.52i)T + (478. - 401. i)T^{2} \) |
| 7 | \( 1 + (-22.6 - 39.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-93.7 + 162. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-94.3 - 112. i)T + (-4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (51.2 - 290. i)T + (-7.84e4 - 2.85e4i)T^{2} \) |
| 23 | \( 1 + (679. + 247. i)T + (2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-623. + 109. i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (260. - 150. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.83e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (891. - 1.06e3i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-149. + 54.3i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (253. + 1.43e3i)T + (-4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 + (473. - 1.30e3i)T + (-6.04e6 - 5.07e6i)T^{2} \) |
| 59 | \( 1 + (3.98e3 + 702. i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (1.85e3 + 674. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (2.64e3 - 467. i)T + (1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-204. - 561. i)T + (-1.94e7 + 1.63e7i)T^{2} \) |
| 73 | \( 1 + (-1.32e3 - 1.10e3i)T + (4.93e6 + 2.79e7i)T^{2} \) |
| 79 | \( 1 + (-2.84e3 + 3.38e3i)T + (-6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (3.40e3 + 5.90e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.09e3 - 2.50e3i)T + (-1.08e7 + 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-1.42e4 - 2.50e3i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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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
−17.83745147291298059546115453850, −16.77567371600738826261500724535, −16.01218078813043306978733189599, −13.83952694863525265628131267932, −11.73574791723766424314666605440, −10.64011321858606172656932218251, −9.267511267315141806944398817848, −8.334259488573669980051490791447, −6.01091303458466124055860214228, −1.77002873895737525550801753212,
1.38745504589128889563510207366, 6.38497883429735167338346533781, 7.52872636426675726457936747438, 9.464536563601231717940862274404, 10.22556725279707533096786653534, 11.82608768969678721614625640144, 14.02110684217979048578493392777, 15.52205630247090766345736828342, 17.07124530908282319863137522161, 17.91326458373241494376788149198
