| 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
−17.54055006765074302670853509847, −16.32190301095735138883777600591, −13.77464894847104477871750352517, −12.76937832014313618664199630462, −11.03985870740908854496273646835, −10.66301024888161675727410380015, −9.032668093955001437486850917994, −6.48889712360946411126568412487, −3.49389117202782567080854544625, −0.68872361649768522889277058069,
5.60409934805501060831662063688, 6.27023333023614057601644083683, 8.493656016027657214833752178486, 9.534637421527875355891612197287, 11.69024145011154862397446499355, 13.44029820670417473757613343155, 15.15274796543280946120307290607, 16.07923551505883652590772128874, 16.91416392438891257228734876123, 18.06373343956747709904734016171
