L(s) = 1 | + (0.984 − 0.173i)2-s + (0.712 + 0.849i)3-s + (0.939 − 0.342i)4-s + (0.849 + 0.712i)6-s + (−4.26 − 2.46i)7-s + (0.866 − 0.5i)8-s + (0.307 − 1.74i)9-s + (−2.20 − 3.82i)11-s + (0.960 + 0.554i)12-s + (1.69 − 2.02i)13-s + (−4.63 − 1.68i)14-s + (0.766 − 0.642i)16-s + (−4.94 + 0.872i)17-s − 1.76i·18-s + (4.21 − 1.11i)19-s + ⋯ |
L(s) = 1 | + (0.696 − 0.122i)2-s + (0.411 + 0.490i)3-s + (0.469 − 0.171i)4-s + (0.346 + 0.291i)6-s + (−1.61 − 0.931i)7-s + (0.306 − 0.176i)8-s + (0.102 − 0.580i)9-s + (−0.665 − 1.15i)11-s + (0.277 + 0.160i)12-s + (0.470 − 0.560i)13-s + (−1.23 − 0.450i)14-s + (0.191 − 0.160i)16-s + (−1.20 + 0.211i)17-s − 0.417i·18-s + (0.966 − 0.256i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0142 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0142 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34678 - 1.36613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34678 - 1.36613i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.21 + 1.11i)T \) |
good | 3 | \( 1 + (-0.712 - 0.849i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (4.26 + 2.46i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.20 + 3.82i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.69 + 2.02i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.94 - 0.872i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.351 + 0.964i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.462 - 2.62i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.01 - 1.76i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.00iT - 37T^{2} \) |
| 41 | \( 1 + (-1.65 + 1.38i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.516 - 1.41i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.76 - 0.310i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.92 - 5.28i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.44 + 13.8i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-13.6 + 4.96i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (13.4 + 2.36i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-14.6 - 5.33i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.10 - 9.66i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.17 + 1.82i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.30 + 3.64i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.72 + 6.48i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (11.7 - 2.06i)T + (91.1 - 33.1i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.908300044259312658215415300536, −9.175955907616734976191669582612, −8.221377141322688501527266334486, −6.98419805814695262098392376388, −6.40632062674301078985642151777, −5.47654706423235505780811466050, −4.17283762206231611703928358380, −3.40434864746548520633783838668, −2.90586418577663891739581681171, −0.64342812765094823923698952421,
2.10997982633525814139780404088, 2.72551401781029483705609796772, 3.92579837772770692467733072614, 5.07389170729324121701587598528, 5.98105255206184221220913747503, 6.87597253562883648935366394353, 7.43981779662673311142003332057, 8.566352373404537275022256879440, 9.421344984241250895218112915642, 10.12974446536093297711152220867