| L(s) = 1 | + (−2.22 + 6.10i)2-s + (−6.80 + 1.19i)3-s + (−20.0 − 16.8i)4-s + (10.4 − 8.75i)5-s + (7.79 − 44.1i)6-s + (−40.5 + 70.1i)7-s + (57.5 − 33.2i)8-s + (−31.2 + 11.3i)9-s + (30.2 + 83.2i)10-s + (22.3 + 38.7i)11-s + (156. + 90.5i)12-s + (275. + 48.6i)13-s + (−338. − 403. i)14-s + (−60.4 + 72.0i)15-s + (2.12 + 12.0i)16-s + (−122. − 44.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 + 1.52i)2-s + (−0.755 + 0.133i)3-s + (−1.25 − 1.05i)4-s + (0.417 − 0.350i)5-s + (0.216 − 1.22i)6-s + (−0.826 + 1.43i)7-s + (0.899 − 0.519i)8-s + (−0.386 + 0.140i)9-s + (0.302 + 0.832i)10-s + (0.184 + 0.319i)11-s + (1.08 + 0.628i)12-s + (1.63 + 0.287i)13-s + (−1.72 − 2.05i)14-s + (−0.268 + 0.320i)15-s + (0.00828 + 0.0469i)16-s + (−0.425 − 0.154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0625894 - 0.558269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0625894 - 0.558269i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-60.7 - 355. i)T \) |
| good | 2 | \( 1 + (2.22 - 6.10i)T + (-12.2 - 10.2i)T^{2} \) |
| 3 | \( 1 + (6.80 - 1.19i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-10.4 + 8.75i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (40.5 - 70.1i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-22.3 - 38.7i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-275. - 48.6i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (122. + 44.7i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-160. - 134. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (493. + 1.35e3i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (179. + 103. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.73e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-788. + 139. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (528. - 443. i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (354. - 128. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.81e3 + 2.16e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (1.65e3 - 4.54e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-1.07e3 - 901. i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (9.83 + 27.0i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-1.84e3 - 2.19e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-516. - 2.92e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-6.81e3 + 1.20e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-646. + 1.12e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-5.23e3 - 922. i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-279. + 767. i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−18.06373343956747709904734016171, −16.91416392438891257228734876123, −16.07923551505883652590772128874, −15.15274796543280946120307290607, −13.44029820670417473757613343155, −11.69024145011154862397446499355, −9.534637421527875355891612197287, −8.493656016027657214833752178486, −6.27023333023614057601644083683, −5.60409934805501060831662063688,
0.68872361649768522889277058069, 3.49389117202782567080854544625, 6.48889712360946411126568412487, 9.032668093955001437486850917994, 10.66301024888161675727410380015, 11.03985870740908854496273646835, 12.76937832014313618664199630462, 13.77464894847104477871750352517, 16.32190301095735138883777600591, 17.54055006765074302670853509847
