L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.09 − 0.629i)5-s + (2.11 − 1.59i)7-s − 0.999i·8-s + (0.629 + 1.09i)10-s + (−1.45 − 2.98i)11-s − 4.08·13-s + (−2.62 + 0.319i)14-s + (−0.5 + 0.866i)16-s + (1.60 + 2.77i)17-s + (3.81 − 6.60i)19-s − 1.25i·20-s + (−0.235 + 3.30i)22-s + (−4.12 + 7.14i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.487 − 0.281i)5-s + (0.799 − 0.600i)7-s − 0.353i·8-s + (0.199 + 0.344i)10-s + (−0.437 − 0.899i)11-s − 1.13·13-s + (−0.701 + 0.0854i)14-s + (−0.125 + 0.216i)16-s + (0.388 + 0.672i)17-s + (0.875 − 1.51i)19-s − 0.281i·20-s + (−0.0502 + 0.705i)22-s + (−0.860 + 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4347669366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4347669366\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.11 + 1.59i)T \) |
| 11 | \( 1 + (1.45 + 2.98i)T \) |
good | 5 | \( 1 + (1.09 + 0.629i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 + (-1.60 - 2.77i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.81 + 6.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.12 - 7.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.54iT - 29T^{2} \) |
| 31 | \( 1 + (7.95 - 4.59i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.154 + 0.266i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 7.57iT - 43T^{2} \) |
| 47 | \( 1 + (4.07 + 2.35i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.39 + 4.15i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.36 - 1.36i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.755 - 1.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.69 + 2.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.50T + 71T^{2} \) |
| 73 | \( 1 + (-0.483 - 0.837i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (13.5 + 7.80i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.32T + 83T^{2} \) |
| 89 | \( 1 + (9.22 + 5.32i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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
−9.227596610080932155180688876827, −8.225425802133091319477101655875, −7.72642380509413811862715587670, −7.12847986986973879976883268921, −5.75288512749189446919996147833, −4.85982276533817976085387232908, −3.88796125832667187454006097254, −2.86128086622444976241157251067, −1.53654168789509208573733717406, −0.21391591246184046653383275121,
1.73322519822277411970392801034, 2.69478422246731129028378557852, 4.13951516434218115309182359407, 5.16055421302930319051954861688, 5.78385832961148256545518657250, 7.12262871029171857860200054863, 7.61931754876390224798897055877, 8.145870567255083249140564273855, 9.248112498528937508399786111093, 9.856237030242829878254630009644