| L(s) = 1 | + (−2.13 + 0.988i)2-s + (0.947 + 0.319i)3-s + (2.29 − 2.70i)4-s + (2.84 + 1.71i)5-s + (−2.34 + 0.254i)6-s + (0.205 − 3.79i)7-s + (−0.972 + 3.50i)8-s + (0.796 + 0.605i)9-s + (−7.78 − 0.846i)10-s + (0.766 − 4.67i)11-s + (3.03 − 1.82i)12-s + (1.32 − 1.00i)13-s + (3.31 + 8.32i)14-s + (2.15 + 2.53i)15-s + (−0.238 − 1.45i)16-s + (0.334 + 6.17i)17-s + ⋯ |
| L(s) = 1 | + (−1.51 + 0.699i)2-s + (0.547 + 0.184i)3-s + (1.14 − 1.35i)4-s + (1.27 + 0.766i)5-s + (−0.955 + 0.103i)6-s + (0.0778 − 1.43i)7-s + (−0.344 + 1.23i)8-s + (0.265 + 0.201i)9-s + (−2.46 − 0.267i)10-s + (0.231 − 1.41i)11-s + (0.876 − 0.527i)12-s + (0.368 − 0.279i)13-s + (0.886 + 2.22i)14-s + (0.555 + 0.654i)15-s + (−0.0596 − 0.363i)16-s + (0.0812 + 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.796934 + 0.268098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.796934 + 0.268098i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.947 - 0.319i)T \) |
| 59 | \( 1 + (-5.49 + 5.36i)T \) |
| good | 2 | \( 1 + (2.13 - 0.988i)T + (1.29 - 1.52i)T^{2} \) |
| 5 | \( 1 + (-2.84 - 1.71i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.205 + 3.79i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (-0.766 + 4.67i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 1.00i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (-0.334 - 6.17i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (0.760 - 0.720i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (6.59 - 1.45i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (-1.47 - 0.682i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (-3.36 - 3.18i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.453 - 1.63i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (5.60 + 1.23i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (0.677 + 4.13i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (11.1 - 6.72i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (3.87 - 0.421i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (0.386 - 0.178i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (4.22 - 15.2i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (-9.90 + 5.95i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (1.94 + 4.88i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (5.10 - 1.71i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (-3.34 + 4.93i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (-9.08 - 4.20i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-1.24 + 3.12i)T + (-70.4 - 66.7i)T^{2} \) |
| 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
−13.26342650064221924166096991238, −11.02901045231584600097474859805, −10.31274684686365679042413683071, −9.942835134253638691582108143697, −8.603153277759226959229490461069, −7.927884819442423485409971013490, −6.63128271795271252679388319343, −6.01254489232387686433811295928, −3.58621913525830404786086646478, −1.53339253753629008087744471750,
1.78310516945601563590333743751, 2.47920233666009031722804972152, 5.02077351944593471664830146006, 6.53726278945850240480390739375, 8.002755719446202843741619355422, 8.897146419764009599749098404241, 9.559249966254478020739805198658, 9.966035245751161706462986298953, 11.70609335404934898242595016627, 12.24096782074528804467828430681
